id	sid	tid	token	lemma	pos
ajrhss-2193	1	1	american	american	PROPN
ajrhss-2193	1	2	journal	journal	PROPN
ajrhss-2193	1	3	of	of	ADP
ajrhss-2193	1	4	research	research	NOUN
ajrhss-2193	1	5	in	in	ADP
ajrhss-2193	1	6	humanities	humanity	NOUN
ajrhss-2193	1	7	and	and	CCONJ
ajrhss-2193	1	8	social	social	ADJ
ajrhss-2193	1	9	sciences	science	NOUN
ajrhss-2193	1	10	issn	issn	PROPN
ajrhss-2193	1	11	(	(	PUNCT
ajrhss-2193	1	12	e	e	NOUN
ajrhss-2193	1	13	):	):	PUNCT
ajrhss-2193	1	14	2832	2832	NUM
ajrhss-2193	1	15	-	-	SYM
ajrhss-2193	1	16	8019	8019	NUM
ajrhss-2193	1	17	volume	volume	NOUN
ajrhss-2193	1	18	25	25	NUM
ajrhss-2193	1	19	,	,	PUNCT
ajrhss-2193	1	20	|	|	ADV
ajrhss-2193	1	21	june	june	PROPN
ajrhss-2193	1	22	2024	2024	NUM
ajrhss-2193	1	23	p	p	NOUN
ajrhss-2193	1	24	a	a	PRON
ajrhss-2193	1	25	g	g	NOUN
ajrhss-2193	1	26	e	e	NOUN
ajrhss-2193	1	27	|	|	ADV
ajrhss-2193	1	28	38	38	NUM
ajrhss-2193	1	29	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2193	1	30	arithmetic	arithmetic	ADJ
ajrhss-2193	1	31	progression	progression	NOUN
ajrhss-2193	1	32	abdurashitov	abdurashitov	PROPN
ajrhss-2193	1	33	rakhmatullo	rakhmatullo	PROPN
ajrhss-2193	1	34	abdukhamitovich	abdukhamitovich	PROPN
ajrhss-2193	1	35	khusenov	khusenov	PROPN
ajrhss-2193	1	36	zikrillo	zikrillo	PROPN
ajrhss-2193	1	37	raxmatillo	raxmatillo	VERB
ajrhss-2193	1	38	ogli	ogli	NOUN
ajrhss-2193	1	39	adkhamov	adkhamov	PROPN
ajrhss-2193	1	40	khasanboy	khasanboy	PROPN
ajrhss-2193	1	41	abdusalom	abdusalom	PROPN
ajrhss-2193	1	42	ogli	ogli	PROPN
ajrhss-2193	1	43	a	a	DET
ajrhss-2193	1	44	b	b	PROPN
ajrhss-2193	1	45	s	s	ADP
ajrhss-2193	1	46	t	t	PROPN
ajrhss-2193	1	47	r	r	NOUN
ajrhss-2193	1	48	a	a	DET
ajrhss-2193	1	49	c	c	NOUN
ajrhss-2193	1	50	t	t	NOUN
ajrhss-2193	1	51	k	k	X
ajrhss-2193	1	52	e	e	PROPN
ajrhss-2193	1	53	y	y	PROPN
ajrhss-2193	1	54	w	w	NOUN
ajrhss-2193	1	55	o	o	NOUN
ajrhss-2193	1	56	r	r	NOUN
ajrhss-2193	1	57	d	d	X
ajrhss-2193	1	58	s	s	VERB
ajrhss-2193	1	59	an	an	DET
ajrhss-2193	1	60	arithmetic	arithmetic	ADJ
ajrhss-2193	1	61	progression	progression	NOUN
ajrhss-2193	1	62	(	(	PUNCT
ajrhss-2193	1	63	ap	ap	PROPN
ajrhss-2193	1	64	)	)	PUNCT
ajrhss-2193	1	65	is	be	AUX
ajrhss-2193	1	66	a	a	DET
ajrhss-2193	1	67	sequence	sequence	NOUN
ajrhss-2193	1	68	of	of	ADP
ajrhss-2193	1	69	numbers	number	NOUN
ajrhss-2193	1	70	where	where	SCONJ
ajrhss-2193	1	71	the	the	DET
ajrhss-2193	1	72	difference	difference	NOUN
ajrhss-2193	1	73	between	between	ADP
ajrhss-2193	1	74	any	any	DET
ajrhss-2193	1	75	two	two	NUM
ajrhss-2193	1	76	consecutive	consecutive	ADJ
ajrhss-2193	1	77	terms	term	NOUN
ajrhss-2193	1	78	is	be	AUX
ajrhss-2193	1	79	constant	constant	ADJ
ajrhss-2193	1	80	,	,	PUNCT
ajrhss-2193	1	81	known	know	VERB
ajrhss-2193	1	82	as	as	ADP
ajrhss-2193	1	83	the	the	DET
ajrhss-2193	1	84	common	common	ADJ
ajrhss-2193	1	85	difference	difference	NOUN
ajrhss-2193	1	86	(	(	PUNCT
ajrhss-2193	1	87	d	d	NOUN
ajrhss-2193	1	88	)	)	PUNCT
ajrhss-2193	1	89	.	.	PUNCT
ajrhss-2193	2	1	the	the	DET
ajrhss-2193	2	2	sequence	sequence	NOUN
ajrhss-2193	2	3	is	be	AUX
ajrhss-2193	2	4	defined	define	VERB
ajrhss-2193	2	5	by	by	ADP
ajrhss-2193	2	6	its	its	PRON
ajrhss-2193	2	7	first	first	ADJ
ajrhss-2193	2	8	term	term	NOUN
ajrhss-2193	2	9	(	(	PUNCT
ajrhss-2193	2	10	a	a	NOUN
ajrhss-2193	2	11	)	)	PUNCT
ajrhss-2193	2	12	and	and	CCONJ
ajrhss-2193	2	13	the	the	DET
ajrhss-2193	2	14	common	common	ADJ
ajrhss-2193	2	15	difference	difference	NOUN
ajrhss-2193	2	16	.	.	PUNCT
ajrhss-2193	3	1	the	the	DET
ajrhss-2193	3	2	general	general	ADJ
ajrhss-2193	3	3	formula	formula	NOUN
ajrhss-2193	3	4	for	for	ADP
ajrhss-2193	3	5	the	the	DET
ajrhss-2193	3	6	\	\	NOUN
ajrhss-2193	3	7	(	(	PUNCT
ajrhss-2193	3	8	n	n	NUM
ajrhss-2193	3	9	\)-th	\)-th	VERB
ajrhss-2193	3	10	term	term	NOUN
ajrhss-2193	3	11	of	of	ADP
ajrhss-2193	3	12	an	an	DET
ajrhss-2193	3	13	ap	ap	PROPN
ajrhss-2193	3	14	is	be	AUX
ajrhss-2193	3	15	\	\	PROPN
ajrhss-2193	3	16	(	(	PUNCT
ajrhss-2193	3	17	a_n	a_n	PROPN
ajrhss-2193	3	18	=	=	PUNCT
ajrhss-2193	3	19	a	a	PRON
ajrhss-2193	3	20	+	+	X
ajrhss-2193	3	21	(	(	PUNCT
ajrhss-2193	3	22	n-1)d	n-1)d	PROPN
ajrhss-2193	3	23	\	\	PROPN
ajrhss-2193	3	24	)	)	PUNCT
ajrhss-2193	3	25	.	.	PUNCT
ajrhss-2193	4	1	the	the	DET
ajrhss-2193	4	2	sum	sum	NOUN
ajrhss-2193	4	3	of	of	ADP
ajrhss-2193	4	4	the	the	DET
ajrhss-2193	4	5	first	first	ADJ
ajrhss-2193	4	6	\	\	PROPN
ajrhss-2193	4	7	(	(	PUNCT
ajrhss-2193	4	8	n	n	PRON
ajrhss-2193	4	9	\	\	ADJ
ajrhss-2193	4	10	)	)	PUNCT
ajrhss-2193	4	11	terms	term	NOUN
ajrhss-2193	4	12	(	(	PUNCT
ajrhss-2193	4	13	\	\	PROPN
ajrhss-2193	4	14	(	(	PUNCT
ajrhss-2193	4	15	s_n	s_n	PROPN
ajrhss-2193	4	16	\	\	PROPN
ajrhss-2193	4	17	)	)	PUNCT
ajrhss-2193	4	18	)	)	PUNCT
ajrhss-2193	4	19	can	can	AUX
ajrhss-2193	4	20	be	be	AUX
ajrhss-2193	4	21	calculated	calculate	VERB
ajrhss-2193	4	22	using	use	VERB
ajrhss-2193	4	23	\	\	PROPN
ajrhss-2193	4	24	(	(	PUNCT
ajrhss-2193	4	25	s_n	s_n	SYM
ajrhss-2193	4	26	=	=	SYM
ajrhss-2193	4	27	\frac{n}{2	\frac{n}{2	PROPN
ajrhss-2193	4	28	}	}	PUNCT
ajrhss-2193	4	29	(	(	PUNCT
ajrhss-2193	4	30	2a	2a	NUM
ajrhss-2193	4	31	+	+	CCONJ
ajrhss-2193	4	32	(	(	PUNCT
ajrhss-2193	4	33	n-1)d	n-1)d	PROPN
ajrhss-2193	4	34	)	)	PUNCT
ajrhss-2193	4	35	\	\	NOUN
ajrhss-2193	4	36	)	)	PUNCT
ajrhss-2193	4	37	or	or	CCONJ
ajrhss-2193	4	38	\	\	PROPN
ajrhss-2193	4	39	(	(	PUNCT
ajrhss-2193	4	40	s_n	s_n	SYM
ajrhss-2193	4	41	=	=	SYM
ajrhss-2193	4	42	\frac{n}{2	\frac{n}{2	PROPN
ajrhss-2193	4	43	}	}	PUNCT
ajrhss-2193	4	44	(	(	PUNCT
ajrhss-2193	4	45	a	a	DET
ajrhss-2193	4	46	+	+	NUM
ajrhss-2193	4	47	l	l	NOUN
ajrhss-2193	4	48	)	)	PUNCT
ajrhss-2193	4	49	\	\	NOUN
ajrhss-2193	4	50	)	)	PUNCT
ajrhss-2193	4	51	,	,	PUNCT
ajrhss-2193	4	52	where	where	SCONJ
ajrhss-2193	4	53	\	\	PROPN
ajrhss-2193	4	54	(	(	PUNCT
ajrhss-2193	4	55	l	l	NOUN
ajrhss-2193	4	56	\	\	PROPN
ajrhss-2193	4	57	)	)	PUNCT
ajrhss-2193	4	58	is	be	AUX
ajrhss-2193	4	59	the	the	DET
ajrhss-2193	4	60	last	last	ADJ
ajrhss-2193	4	61	term	term	NOUN
ajrhss-2193	4	62	.	.	PUNCT
ajrhss-2193	5	1	aps	ap	NOUN
ajrhss-2193	5	2	are	be	AUX
ajrhss-2193	5	3	widely	widely	ADV
ajrhss-2193	5	4	used	use	VERB
ajrhss-2193	5	5	in	in	ADP
ajrhss-2193	5	6	various	various	ADJ
ajrhss-2193	5	7	fields	field	NOUN
ajrhss-2193	5	8	such	such	ADJ
ajrhss-2193	5	9	as	as	ADP
ajrhss-2193	5	10	finance	finance	NOUN
ajrhss-2193	5	11	,	,	PUNCT
ajrhss-2193	5	12	physics	physics	NOUN
ajrhss-2193	5	13	,	,	PUNCT
ajrhss-2193	5	14	engineering	engineering	NOUN
ajrhss-2193	5	15	,	,	PUNCT
ajrhss-2193	5	16	and	and	CCONJ
ajrhss-2193	5	17	computer	computer	NOUN
ajrhss-2193	5	18	science	science	NOUN
ajrhss-2193	5	19	,	,	PUNCT
ajrhss-2193	5	20	due	due	ADP
ajrhss-2193	5	21	to	to	ADP
ajrhss-2193	5	22	their	their	PRON
ajrhss-2193	5	23	regular	regular	ADJ
ajrhss-2193	5	24	and	and	CCONJ
ajrhss-2193	5	25	predictable	predictable	ADJ
ajrhss-2193	5	26	nature	nature	NOUN
ajrhss-2193	5	27	.	.	PUNCT
ajrhss-2193	6	1	arithmetic	arithmetic	ADJ
ajrhss-2193	6	2	progression	progression	NOUN
ajrhss-2193	6	3	,	,	PUNCT
ajrhss-2193	6	4	common	common	ADJ
ajrhss-2193	6	5	difference	difference	NOUN
ajrhss-2193	6	6	,	,	PUNCT
ajrhss-2193	6	7	first	first	ADJ
ajrhss-2193	6	8	term	term	NOUN
ajrhss-2193	6	9	,	,	PUNCT
ajrhss-2193	6	10	(	(	PUNCT
ajrhss-2193	6	11	n)-th	n)-th	PROPN
ajrhss-2193	6	12	term(a_n	term(a_n	NUM
ajrhss-2193	6	13	)	)	PUNCT
ajrhss-2193	6	14	,	,	PUNCT
ajrhss-2193	6	15	sequence	sequence	NOUN
ajrhss-2193	6	16	,	,	PUNCT
ajrhss-2193	6	17	linear	linear	ADJ
ajrhss-2193	6	18	sequence	sequence	NOUN
ajrhss-2193	6	19	,	,	PUNCT
ajrhss-2193	6	20	uniform	uniform	ADJ
ajrhss-2193	6	21	difference	difference	NOUN
ajrhss-2193	6	22	,	,	PUNCT
ajrhss-2193	6	23	incremental	incremental	ADJ
ajrhss-2193	6	24	change	change	NOUN
ajrhss-2193	6	25	,	,	PUNCT
ajrhss-2193	6	26	mathematical	mathematical	ADJ
ajrhss-2193	6	27	series	series	NOUN
ajrhss-2193	6	28	,	,	PUNCT
ajrhss-2193	6	29	geometric	geometric	ADJ
ajrhss-2193	6	30	progression	progression	NOUN
ajrhss-2193	6	31	,	,	PUNCT
ajrhss-2193	6	32	algorithm	algorithm	NOUN
ajrhss-2193	6	33	,	,	PUNCT
ajrhss-2193	6	34	signal	signal	ADJ
ajrhss-2193	6	35	processing	processing	NOUN
ajrhss-2193	6	36	.	.	PUNCT
ajrhss-2193	7	1	introduction	introduction	NOUN
ajrhss-2193	7	2	an	an	DET
ajrhss-2193	7	3	arithmetic	arithmetic	ADJ
ajrhss-2193	7	4	progression	progression	NOUN
ajrhss-2193	7	5	(	(	PUNCT
ajrhss-2193	7	6	ap	ap	PROPN
ajrhss-2193	7	7	)	)	PUNCT
ajrhss-2193	7	8	is	be	AUX
ajrhss-2193	7	9	a	a	DET
ajrhss-2193	7	10	sequence	sequence	NOUN
ajrhss-2193	7	11	of	of	ADP
ajrhss-2193	7	12	numbers	number	NOUN
ajrhss-2193	7	13	in	in	ADP
ajrhss-2193	7	14	which	which	PRON
ajrhss-2193	7	15	the	the	DET
ajrhss-2193	7	16	difference	difference	NOUN
ajrhss-2193	7	17	between	between	ADP
ajrhss-2193	7	18	any	any	DET
ajrhss-2193	7	19	two	two	NUM
ajrhss-2193	7	20	consecutive	consecutive	ADJ
ajrhss-2193	7	21	terms	term	NOUN
ajrhss-2193	7	22	is	be	AUX
ajrhss-2193	7	23	constant	constant	ADJ
ajrhss-2193	7	24	.	.	PUNCT
ajrhss-2193	8	1	this	this	DET
ajrhss-2193	8	2	fixed	fix	VERB
ajrhss-2193	8	3	difference	difference	NOUN
ajrhss-2193	8	4	is	be	AUX
ajrhss-2193	8	5	called	call	VERB
ajrhss-2193	8	6	the	the	DET
ajrhss-2193	8	7	"	"	PUNCT
ajrhss-2193	8	8	common	common	ADJ
ajrhss-2193	8	9	difference	difference	NOUN
ajrhss-2193	8	10	,	,	PUNCT
ajrhss-2193	8	11	"	"	PUNCT
ajrhss-2193	8	12	denoted	denote	VERB
ajrhss-2193	8	13	by	by	ADP
ajrhss-2193	8	14	\	\	PROPN
ajrhss-2193	8	15	(	(	PUNCT
ajrhss-2193	8	16	d	d	NOUN
ajrhss-2193	8	17	\	\	PROPN
ajrhss-2193	8	18	)	)	PUNCT
ajrhss-2193	8	19	.	.	PUNCT
ajrhss-2193	9	1	key	key	ADJ
ajrhss-2193	9	2	features	feature	NOUN
ajrhss-2193	9	3	of	of	ADP
ajrhss-2193	9	4	an	an	DET
ajrhss-2193	9	5	arithmetic	arithmetic	ADJ
ajrhss-2193	9	6	progression	progression	NOUN
ajrhss-2193	9	7	1	1	NUM
ajrhss-2193	9	8	.	.	PUNCT
ajrhss-2193	9	9	first	first	ADJ
ajrhss-2193	9	10	term	term	NOUN
ajrhss-2193	9	11	(	(	PUNCT
ajrhss-2193	9	12	\	\	PROPN
ajrhss-2193	9	13	(	(	PUNCT
ajrhss-2193	9	14	a	a	DET
ajrhss-2193	9	15	\	\	NOUN
ajrhss-2193	9	16	)	)	PUNCT
ajrhss-2193	9	17	):	):	PUNCT
ajrhss-2193	9	18	the	the	DET
ajrhss-2193	9	19	starting	starting	NOUN
ajrhss-2193	9	20	point	point	NOUN
ajrhss-2193	9	21	of	of	ADP
ajrhss-2193	9	22	the	the	DET
ajrhss-2193	9	23	sequence	sequence	NOUN
ajrhss-2193	9	24	.	.	PUNCT
ajrhss-2193	10	1	2	2	X
ajrhss-2193	10	2	.	.	X
ajrhss-2193	10	3	common	common	ADJ
ajrhss-2193	10	4	difference	difference	NOUN
ajrhss-2193	10	5	(	(	PUNCT
ajrhss-2193	10	6	\	\	PROPN
ajrhss-2193	10	7	(	(	PUNCT
ajrhss-2193	10	8	d	d	PROPN
ajrhss-2193	10	9	\	\	PROPN
ajrhss-2193	10	10	)	)	PUNCT
ajrhss-2193	10	11	):	):	PUNCT
ajrhss-2193	10	12	the	the	DET
ajrhss-2193	10	13	constant	constant	ADJ
ajrhss-2193	10	14	amount	amount	NOUN
ajrhss-2193	10	15	added	add	VERB
ajrhss-2193	10	16	(	(	PUNCT
ajrhss-2193	10	17	or	or	CCONJ
ajrhss-2193	10	18	subtracted	subtract	VERB
ajrhss-2193	10	19	)	)	PUNCT
ajrhss-2193	10	20	to	to	ADP
ajrhss-2193	10	21	each	each	DET
ajrhss-2193	10	22	term	term	NOUN
ajrhss-2193	10	23	to	to	PART
ajrhss-2193	10	24	get	get	VERB
ajrhss-2193	10	25	the	the	DET
ajrhss-2193	10	26	next	next	ADJ
ajrhss-2193	10	27	term	term	NOUN
ajrhss-2193	10	28	.	.	PUNCT
ajrhss-2193	11	1	general	general	ADJ
ajrhss-2193	11	2	form	form	VERB
ajrhss-2193	11	3	the	the	DET
ajrhss-2193	11	4	sequence	sequence	NOUN
ajrhss-2193	11	5	can	can	AUX
ajrhss-2193	11	6	be	be	AUX
ajrhss-2193	11	7	written	write	VERB
ajrhss-2193	11	8	as	as	ADP
ajrhss-2193	11	9	:	:	PUNCT
ajrhss-2193	11	10	a	a	X
ajrhss-2193	11	11	,	,	PUNCT
ajrhss-2193	11	12	a	a	PRON
ajrhss-2193	11	13	+	+	X
ajrhss-2193	11	14	d	d	PROPN
ajrhss-2193	11	15	,	,	PUNCT
ajrhss-2193	11	16	a	a	DET
ajrhss-2193	11	17	+	+	X
ajrhss-2193	11	18	2d	2d	NOUN
ajrhss-2193	11	19	,	,	PUNCT
ajrhss-2193	11	20	a	a	DET
ajrhss-2193	11	21	+	+	X
ajrhss-2193	11	22	3d	3d	NUM
ajrhss-2193	11	23	,	,	PUNCT
ajrhss-2193	11	24	....	....	PUNCT
ajrhss-2193	12	1	nth	nth	NOUN
ajrhss-2193	12	2	term	term	NOUN
ajrhss-2193	12	3	formula	formula	VERB
ajrhss-2193	12	4	the	the	DET
ajrhss-2193	12	5	\	\	NOUN
ajrhss-2193	12	6	(	(	PUNCT
ajrhss-2193	12	7	n	n	PRON
ajrhss-2193	12	8	\)th	\)th	NOUN
ajrhss-2193	12	9	term	term	NOUN
ajrhss-2193	12	10	(	(	PUNCT
ajrhss-2193	12	11	\	\	PROPN
ajrhss-2193	12	12	(	(	PUNCT
ajrhss-2193	12	13	t_n	t_n	PUNCT
ajrhss-2193	12	14	\	\	NOUN
ajrhss-2193	12	15	)	)	PUNCT
ajrhss-2193	12	16	)	)	PUNCT
ajrhss-2193	12	17	of	of	ADP
ajrhss-2193	12	18	an	an	DET
ajrhss-2193	12	19	arithmetic	arithmetic	ADJ
ajrhss-2193	12	20	progression	progression	NOUN
ajrhss-2193	12	21	is	be	AUX
ajrhss-2193	12	22	given	give	VERB
ajrhss-2193	12	23	by:\	by:\	NOUN
ajrhss-2193	12	24	[	[	PUNCT
ajrhss-2193	12	25	t_n	t_n	PUNCT
ajrhss-2193	12	26	=	=	PUNCT
ajrhss-2193	13	1	a	a	PRON
ajrhss-2193	13	2	+	+	X
ajrhss-2193	14	1	(	(	PUNCT
ajrhss-2193	14	2	n	n	NUM
ajrhss-2193	14	3	1)d	1)d	NUM
ajrhss-2193	14	4	\	\	NOUN
ajrhss-2193	14	5	]	]	PUNCT
ajrhss-2193	14	6	sum	sum	NOUN
ajrhss-2193	14	7	of	of	ADP
ajrhss-2193	14	8	the	the	DET
ajrhss-2193	14	9	first	first	ADJ
ajrhss-2193	14	10	n	n	PART
ajrhss-2193	14	11	terms	term	NOUN
ajrhss-2193	14	12	the	the	DET
ajrhss-2193	14	13	sum	sum	NOUN
ajrhss-2193	14	14	of	of	ADP
ajrhss-2193	14	15	the	the	DET
ajrhss-2193	14	16	first	first	ADJ
ajrhss-2193	14	17	\	\	PROPN
ajrhss-2193	14	18	(	(	PUNCT
ajrhss-2193	14	19	n	n	PRON
ajrhss-2193	14	20	\	\	ADJ
ajrhss-2193	14	21	)	)	PUNCT
ajrhss-2193	14	22	terms	term	NOUN
ajrhss-2193	14	23	(	(	PUNCT
ajrhss-2193	14	24	\	\	PROPN
ajrhss-2193	14	25	(	(	PUNCT
ajrhss-2193	14	26	s_n	s_n	PROPN
ajrhss-2193	14	27	\	\	PROPN
ajrhss-2193	14	28	)	)	PUNCT
ajrhss-2193	14	29	)	)	PUNCT
ajrhss-2193	14	30	of	of	ADP
ajrhss-2193	14	31	an	an	DET
ajrhss-2193	14	32	arithmetic	arithmetic	ADJ
ajrhss-2193	14	33	progression	progression	NOUN
ajrhss-2193	14	34	is	be	AUX
ajrhss-2193	14	35	:	:	PUNCT
ajrhss-2193	14	36	sn	sn	PROPN
ajrhss-2193	14	37	=	=	SYM
ajrhss-2193	14	38	\frac{n}{2	\frac{n}{2	PROPN
ajrhss-2193	14	39	}	}	PUNCT
ajrhss-2193	14	40	(	(	PUNCT
ajrhss-2193	14	41	2a	2a	NUM
ajrhss-2193	14	42	+	+	CCONJ
ajrhss-2193	14	43	(	(	PUNCT
ajrhss-2193	14	44	n	n	PRON
ajrhss-2193	14	45	1)d)or	1)d)or	PROPN
ajrhss-2193	14	46	equivalently	equivalently	ADV
ajrhss-2193	14	47	,	,	PUNCT
ajrhss-2193	14	48	sn	sn	PROPN
ajrhss-2193	14	49	=	=	SYM
ajrhss-2193	14	50	\frac{n}{2	\frac{n}{2	PROPN
ajrhss-2193	14	51	}	}	PUNCT
ajrhss-2193	14	52	(	(	PUNCT
ajrhss-2193	14	53	a	a	DET
ajrhss-2193	14	54	+	+	X
ajrhss-2193	14	55	l	l	NOUN
ajrhss-2193	14	56	)	)	PUNCT
ajrhss-2193	14	57	where	where	SCONJ
ajrhss-2193	14	58	\	\	PROPN
ajrhss-2193	14	59	(	(	PUNCT
ajrhss-2193	14	60	l	l	NOUN
ajrhss-2193	14	61	\	\	PROPN
ajrhss-2193	14	62	)	)	PUNCT
ajrhss-2193	14	63	is	be	AUX
ajrhss-2193	14	64	the	the	DET
ajrhss-2193	14	65	last	last	ADJ
ajrhss-2193	14	66	term	term	NOUN
ajrhss-2193	14	67	of	of	ADP
ajrhss-2193	14	68	the	the	DET
ajrhss-2193	14	69	sequence	sequence	NOUN
ajrhss-2193	14	70	.	.	PUNCT
ajrhss-2193	15	1	example	example	NOUN
ajrhss-2193	15	2	consider	consider	VERB
ajrhss-2193	15	3	the	the	DET
ajrhss-2193	15	4	arithmetic	arithmetic	ADJ
ajrhss-2193	15	5	progression	progression	NOUN
ajrhss-2193	15	6	:	:	PUNCT
ajrhss-2193	15	7	3	3	NUM
ajrhss-2193	15	8	,	,	PUNCT
ajrhss-2193	15	9	7	7	NUM
ajrhss-2193	15	10	,	,	PUNCT
ajrhss-2193	15	11	11	11	NUM
ajrhss-2193	15	12	,	,	PUNCT
ajrhss-2193	15	13	15	15	NUM
ajrhss-2193	15	14	,	,	PUNCT
ajrhss-2193	15	15	...	...	PUNCT
ajrhss-2193	15	16	american	american	ADJ
ajrhss-2193	15	17	journal	journal	PROPN
ajrhss-2193	15	18	of	of	ADP
ajrhss-2193	15	19	research	research	NOUN
ajrhss-2193	15	20	in	in	ADP
ajrhss-2193	15	21	humanities	humanity	NOUN
ajrhss-2193	15	22	and	and	CCONJ
ajrhss-2193	15	23	social	social	ADJ
ajrhss-2193	15	24	sciences	science	NOUN
ajrhss-2193	15	25	volume	volume	NOUN
ajrhss-2193	15	26	25	25	NUM
ajrhss-2193	15	27	june	june	PROPN
ajrhss-2193	15	28	2024	2024	NUM
ajrhss-2193	15	29	p	p	NOUN
ajrhss-2193	15	30	a	a	DET
ajrhss-2193	15	31	g	g	NOUN
ajrhss-2193	15	32	e	e	NOUN
ajrhss-2193	15	33	|	|	ADV
ajrhss-2193	15	34	39	39	NUM
ajrhss-2193	15	35	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2193	15	36	here	here	ADV
ajrhss-2193	15	37	,	,	PUNCT
ajrhss-2193	15	38	the	the	DET
ajrhss-2193	15	39	first	first	ADJ
ajrhss-2193	15	40	term	term	NOUN
ajrhss-2193	15	41	\	\	PROPN
ajrhss-2193	15	42	(	(	PUNCT
ajrhss-2193	15	43	a	a	DET
ajrhss-2193	15	44	=	=	SYM
ajrhss-2193	15	45	3	3	NUM
ajrhss-2193	15	46	\	\	NOUN
ajrhss-2193	15	47	)	)	PUNCT
ajrhss-2193	15	48	.	.	PUNCT
ajrhss-2193	16	1	the	the	DET
ajrhss-2193	16	2	common	common	ADJ
ajrhss-2193	16	3	difference	difference	NOUN
ajrhss-2193	16	4	\	\	PROPN
ajrhss-2193	16	5	(	(	PUNCT
ajrhss-2193	16	6	d	d	NOUN
ajrhss-2193	16	7	=	=	SYM
ajrhss-2193	16	8	4	4	NUM
ajrhss-2193	16	9	\	\	NOUN
ajrhss-2193	16	10	)	)	PUNCT
ajrhss-2193	16	11	.	.	PUNCT
ajrhss-2193	17	1	to	to	PART
ajrhss-2193	17	2	find	find	VERB
ajrhss-2193	17	3	the	the	DET
ajrhss-2193	17	4	5th	5th	ADJ
ajrhss-2193	17	5	term	term	NOUN
ajrhss-2193	17	6	(	(	PUNCT
ajrhss-2193	17	7	\	\	PROPN
ajrhss-2193	17	8	(	(	PUNCT
ajrhss-2193	17	9	t_5	t_5	PROPN
ajrhss-2193	17	10	\	\	PROPN
ajrhss-2193	17	11	)	)	PUNCT
ajrhss-2193	17	12	):	):	PUNCT
ajrhss-2193	18	1	\	\	PROPN
ajrhss-2193	18	2	[	[	PUNCT
ajrhss-2193	18	3	t_5	t_5	X
ajrhss-2193	18	4	=	=	PUNCT
ajrhss-2193	18	5	3	3	NUM
ajrhss-2193	18	6	+	+	CCONJ
ajrhss-2193	18	7	(	(	PUNCT
ajrhss-2193	18	8	5	5	NUM
ajrhss-2193	18	9	1	1	NUM
ajrhss-2193	18	10	)	)	PUNCT
ajrhss-2193	18	11	\cdot	\cdot	NOUN
ajrhss-2193	18	12	4	4	NUM
ajrhss-2193	18	13	=	=	SYM
ajrhss-2193	18	14	3	3	NUM
ajrhss-2193	18	15	+	+	NUM
ajrhss-2193	18	16	16	16	NUM
ajrhss-2193	18	17	=	=	SYM
ajrhss-2193	18	18	19	19	NUM
ajrhss-2193	18	19	\	\	NOUN
ajrhss-2193	18	20	]	]	PUNCT
ajrhss-2193	18	21	to	to	PART
ajrhss-2193	18	22	find	find	VERB
ajrhss-2193	18	23	the	the	DET
ajrhss-2193	18	24	sum	sum	NOUN
ajrhss-2193	18	25	of	of	ADP
ajrhss-2193	18	26	the	the	DET
ajrhss-2193	18	27	first	first	ADJ
ajrhss-2193	18	28	5	5	NUM
ajrhss-2193	18	29	terms	term	NOUN
ajrhss-2193	18	30	(	(	PUNCT
ajrhss-2193	18	31	\	\	PROPN
ajrhss-2193	18	32	(	(	PUNCT
ajrhss-2193	18	33	s_5	s_5	NOUN
ajrhss-2193	18	34	\	\	PROPN
ajrhss-2193	18	35	)	)	PUNCT
ajrhss-2193	18	36	):	):	PUNCT
ajrhss-2193	18	37	\	\	PROPN
ajrhss-2193	18	38	[	[	PUNCT
ajrhss-2193	18	39	s_5	s_5	X
ajrhss-2193	18	40	=	=	SYM
ajrhss-2193	18	41	\frac{5}{2	\frac{5}{2	PROPN
ajrhss-2193	18	42	}	}	PUNCT
ajrhss-2193	18	43	\left(2	\left(2	NOUN
ajrhss-2193	18	44	\cdot	\cdot	NOUN
ajrhss-2193	18	45	3	3	NUM
ajrhss-2193	18	46	+	+	CCONJ
ajrhss-2193	18	47	(	(	PUNCT
ajrhss-2193	18	48	5	5	NUM
ajrhss-2193	18	49	1	1	NUM
ajrhss-2193	18	50	)	)	PUNCT
ajrhss-2193	18	51	\cdot	\cdot	NOUN
ajrhss-2193	18	52	4\right	4\right	NUM
ajrhss-2193	18	53	)	)	PUNCT
ajrhss-2193	18	54	=	=	SYM
ajrhss-2193	18	55	\frac{5}{2	\frac{5}{2	PROPN
ajrhss-2193	18	56	}	}	PUNCT
ajrhss-2193	18	57	(	(	PUNCT
ajrhss-2193	18	58	6	6	NUM
ajrhss-2193	18	59	+	+	SYM
ajrhss-2193	18	60	16	16	NUM
ajrhss-2193	18	61	)	)	PUNCT
ajrhss-2193	18	62	=	=	SYM
ajrhss-2193	18	63	\frac{5}{2	\frac{5}{2	PROPN
ajrhss-2193	18	64	}	}	PUNCT
ajrhss-2193	18	65	\cdot	\cdot	NOUN
ajrhss-2193	18	66	22	22	NUM
ajrhss-2193	18	67	=	=	SYM
ajrhss-2193	18	68	55	55	NUM
ajrhss-2193	18	69	\	\	NOUN
ajrhss-2193	18	70	]	]	PUNCT
ajrhss-2193	18	71	applications	application	NOUN
ajrhss-2193	18	72	arithmetic	arithmetic	ADJ
ajrhss-2193	18	73	progressions	progression	NOUN
ajrhss-2193	18	74	are	be	AUX
ajrhss-2193	18	75	used	use	VERB
ajrhss-2193	18	76	in	in	ADP
ajrhss-2193	18	77	various	various	ADJ
ajrhss-2193	18	78	fields	field	NOUN
ajrhss-2193	18	79	,	,	PUNCT
ajrhss-2193	18	80	such	such	ADJ
ajrhss-2193	18	81	as	as	ADP
ajrhss-2193	18	82	:	:	PUNCT
ajrhss-2193	18	83	finance	finance	NOUN
ajrhss-2193	18	84	:	:	PUNCT
ajrhss-2193	18	85	calculating	calculate	VERB
ajrhss-2193	18	86	loan	loan	NOUN
ajrhss-2193	18	87	payments	payment	NOUN
ajrhss-2193	18	88	and	and	CCONJ
ajrhss-2193	18	89	savings	saving	NOUN
ajrhss-2193	18	90	.	.	PUNCT
ajrhss-2193	19	1	computer	computer	NOUN
ajrhss-2193	19	2	science	science	PROPN
ajrhss-2193	19	3	:	:	PUNCT
ajrhss-2193	19	4	loop	loop	NOUN
ajrhss-2193	19	5	iterations	iteration	NOUN
ajrhss-2193	19	6	and	and	CCONJ
ajrhss-2193	19	7	algorithm	algorithm	NOUN
ajrhss-2193	19	8	design	design	NOUN
ajrhss-2193	19	9	.	.	PUNCT
ajrhss-2193	20	1	physics	physics	NOUN
ajrhss-2193	20	2	:	:	PUNCT
ajrhss-2193	20	3	describing	describe	VERB
ajrhss-2193	20	4	uniform	uniform	ADJ
ajrhss-2193	20	5	motion	motion	NOUN
ajrhss-2193	20	6	.	.	PUNCT
ajrhss-2193	21	1	conclusion	conclusion	NOUN
ajrhss-2193	21	2	arithmetic	arithmetic	ADJ
ajrhss-2193	21	3	progressions	progression	NOUN
ajrhss-2193	21	4	are	be	AUX
ajrhss-2193	21	5	simple	simple	ADJ
ajrhss-2193	21	6	yet	yet	CCONJ
ajrhss-2193	21	7	powerful	powerful	ADJ
ajrhss-2193	21	8	mathematical	mathematical	ADJ
ajrhss-2193	21	9	sequences	sequence	NOUN
ajrhss-2193	21	10	characterized	characterize	VERB
ajrhss-2193	21	11	by	by	ADP
ajrhss-2193	21	12	a	a	DET
ajrhss-2193	21	13	constant	constant	ADJ
ajrhss-2193	21	14	difference	difference	NOUN
ajrhss-2193	21	15	between	between	ADP
ajrhss-2193	21	16	terms	term	NOUN
ajrhss-2193	21	17	.	.	PUNCT
ajrhss-2193	22	1	their	their	PRON
ajrhss-2193	22	2	straightforward	straightforward	ADJ
ajrhss-2193	22	3	nature	nature	NOUN
ajrhss-2193	22	4	makes	make	VERB
ajrhss-2193	22	5	them	they	PRON
ajrhss-2193	22	6	useful	useful	ADJ
ajrhss-2193	22	7	for	for	ADP
ajrhss-2193	22	8	solving	solve	VERB
ajrhss-2193	22	9	practical	practical	ADJ
ajrhss-2193	22	10	problems	problem	NOUN
ajrhss-2193	22	11	in	in	ADP
ajrhss-2193	22	12	diverse	diverse	ADJ
ajrhss-2193	22	13	fields	field	NOUN
ajrhss-2193	22	14	.	.	PUNCT
ajrhss-2193	23	1	here	here	ADV
ajrhss-2193	23	2	are	be	AUX
ajrhss-2193	23	3	some	some	DET
ajrhss-2193	23	4	more	more	ADV
ajrhss-2193	23	5	detailed	detailed	ADJ
ajrhss-2193	23	6	aspects	aspect	NOUN
ajrhss-2193	23	7	and	and	CCONJ
ajrhss-2193	23	8	applications	application	NOUN
ajrhss-2193	23	9	of	of	ADP
ajrhss-2193	23	10	arithmetic	arithmetic	ADJ
ajrhss-2193	23	11	progressions	progression	NOUN
ajrhss-2193	23	12	(	(	PUNCT
ajrhss-2193	23	13	ap	ap	INTJ
ajrhss-2193	23	14	):	):	PUNCT
ajrhss-2193	23	15	properties	property	NOUN
ajrhss-2193	23	16	of	of	ADP
ajrhss-2193	23	17	arithmetic	arithmetic	ADJ
ajrhss-2193	23	18	progression	progression	NOUN
ajrhss-2193	23	19	1	1	NUM
ajrhss-2193	23	20	.	.	PUNCT
ajrhss-2193	23	21	common	common	ADJ
ajrhss-2193	23	22	difference	difference	NOUN
ajrhss-2193	23	23	(	(	PUNCT
ajrhss-2193	23	24	d	d	NOUN
ajrhss-2193	23	25	):	):	PUNCT
ajrhss-2193	23	26	if	if	SCONJ
ajrhss-2193	23	27	the	the	DET
ajrhss-2193	23	28	common	common	ADJ
ajrhss-2193	23	29	difference	difference	NOUN
ajrhss-2193	23	30	\	\	PROPN
ajrhss-2193	23	31	(	(	PUNCT
ajrhss-2193	23	32	d	d	PROPN
ajrhss-2193	23	33	\	\	PROPN
ajrhss-2193	23	34	)	)	PUNCT
ajrhss-2193	23	35	is	be	AUX
ajrhss-2193	23	36	positive	positive	ADJ
ajrhss-2193	23	37	,	,	PUNCT
ajrhss-2193	23	38	the	the	DET
ajrhss-2193	23	39	sequence	sequence	NOUN
ajrhss-2193	23	40	is	be	AUX
ajrhss-2193	23	41	increasing	increase	VERB
ajrhss-2193	23	42	.	.	PUNCT
ajrhss-2193	24	1	if	if	SCONJ
ajrhss-2193	24	2	\	\	PROPN
ajrhss-2193	24	3	(	(	PUNCT
ajrhss-2193	24	4	d	d	PROPN
ajrhss-2193	24	5	\	\	PROPN
ajrhss-2193	24	6	)	)	PUNCT
ajrhss-2193	24	7	is	be	AUX
ajrhss-2193	24	8	negative	negative	ADJ
ajrhss-2193	24	9	,	,	PUNCT
ajrhss-2193	24	10	the	the	DET
ajrhss-2193	24	11	sequence	sequence	NOUN
ajrhss-2193	24	12	is	be	AUX
ajrhss-2193	24	13	decreasing	decrease	VERB
ajrhss-2193	24	14	.	.	PUNCT
ajrhss-2193	25	1	if	if	SCONJ
ajrhss-2193	25	2	\	\	PROPN
ajrhss-2193	25	3	(	(	PUNCT
ajrhss-2193	25	4	d	d	PROPN
ajrhss-2193	25	5	\	\	PROPN
ajrhss-2193	25	6	)	)	PUNCT
ajrhss-2193	25	7	is	be	AUX
ajrhss-2193	25	8	zero	zero	NUM
ajrhss-2193	25	9	,	,	PUNCT
ajrhss-2193	25	10	all	all	DET
ajrhss-2193	25	11	terms	term	NOUN
ajrhss-2193	25	12	of	of	ADP
ajrhss-2193	25	13	the	the	DET
ajrhss-2193	25	14	sequence	sequence	NOUN
ajrhss-2193	25	15	are	be	AUX
ajrhss-2193	25	16	the	the	DET
ajrhss-2193	25	17	same	same	ADJ
ajrhss-2193	25	18	.	.	PUNCT
ajrhss-2193	26	1	2	2	X
ajrhss-2193	26	2	.	.	PUNCT
ajrhss-2193	26	3	mean	mean	ADJ
ajrhss-2193	26	4	property	property	NOUN
ajrhss-2193	26	5	:	:	PUNCT
ajrhss-2193	26	6	in	in	ADP
ajrhss-2193	26	7	an	an	DET
ajrhss-2193	26	8	arithmetic	arithmetic	ADJ
ajrhss-2193	26	9	progression	progression	NOUN
ajrhss-2193	26	10	,	,	PUNCT
ajrhss-2193	26	11	the	the	DET
ajrhss-2193	26	12	term	term	NOUN
ajrhss-2193	26	13	in	in	ADP
ajrhss-2193	26	14	the	the	DET
ajrhss-2193	26	15	middle	middle	NOUN
ajrhss-2193	26	16	of	of	ADP
ajrhss-2193	26	17	any	any	DET
ajrhss-2193	26	18	three	three	NUM
ajrhss-2193	26	19	consecutive	consecutive	ADJ
ajrhss-2193	26	20	terms	term	NOUN
ajrhss-2193	26	21	is	be	AUX
ajrhss-2193	26	22	the	the	DET
ajrhss-2193	26	23	average	average	NOUN
ajrhss-2193	26	24	of	of	ADP
ajrhss-2193	26	25	the	the	DET
ajrhss-2193	26	26	other	other	ADJ
ajrhss-2193	26	27	two	two	NUM
ajrhss-2193	26	28	.	.	PUNCT
ajrhss-2193	27	1	mathematically	mathematically	ADV
ajrhss-2193	27	2	,	,	PUNCT
ajrhss-2193	27	3	for	for	ADP
ajrhss-2193	27	4	any	any	DET
ajrhss-2193	27	5	three	three	NUM
ajrhss-2193	27	6	consecutive	consecutive	ADJ
ajrhss-2193	27	7	terms	term	NOUN
ajrhss-2193	27	8	\	\	PROPN
ajrhss-2193	27	9	(	(	PUNCT
ajrhss-2193	27	10	a_{n-1	a_{n-1	NOUN
ajrhss-2193	27	11	}	}	PUNCT
ajrhss-2193	27	12	,	,	PUNCT
ajrhss-2193	27	13	a_n	a_n	NUM
ajrhss-2193	27	14	,	,	PUNCT
ajrhss-2193	27	15	a_{n+1	a_{n+1	NOUN
ajrhss-2193	27	16	}	}	PUNCT
ajrhss-2193	27	17	\	\	NOUN
ajrhss-2193	27	18	):	):	PUNCT
ajrhss-2193	27	19	\	\	PROPN
ajrhss-2193	27	20	[	[	PUNCT
ajrhss-2193	27	21	a_n	a_n	PROPN
ajrhss-2193	27	22	=	=	SYM
ajrhss-2193	27	23	\frac{a_{n-1	\frac{a_{n-1	NOUN
ajrhss-2193	27	24	}	}	PUNCT
ajrhss-2193	27	25	+	+	NUM
ajrhss-2193	27	26	a_{n+1}}{2	a_{n+1}}{2	NOUN
ajrhss-2193	27	27	}	}	PUNCT
ajrhss-2193	27	28	\	\	NOUN
ajrhss-2193	27	29	]	]	PUNCT
ajrhss-2193	27	30	deriving	derive	VERB
ajrhss-2193	27	31	formulas	formula	NOUN
ajrhss-2193	27	32	\	\	PROPN
ajrhss-2193	27	33	(	(	PUNCT
ajrhss-2193	27	34	n	n	NUM
ajrhss-2193	27	35	\)-th	\)-th	NOUN
ajrhss-2193	27	36	term	term	NOUN
ajrhss-2193	27	37	formula	formula	NOUN
ajrhss-2193	27	38	the	the	DET
ajrhss-2193	27	39	\	\	NOUN
ajrhss-2193	27	40	(	(	PUNCT
ajrhss-2193	27	41	n	n	NUM
ajrhss-2193	27	42	\)-th	\)-th	NOUN
ajrhss-2193	27	43	term	term	NOUN
ajrhss-2193	27	44	\	\	PROPN
ajrhss-2193	27	45	(	(	PUNCT
ajrhss-2193	27	46	a_n	a_n	NUM
ajrhss-2193	27	47	\	\	NOUN
ajrhss-2193	27	48	)	)	PUNCT
ajrhss-2193	27	49	of	of	ADP
ajrhss-2193	27	50	an	an	DET
ajrhss-2193	27	51	ap	ap	PROPN
ajrhss-2193	27	52	can	can	AUX
ajrhss-2193	27	53	be	be	AUX
ajrhss-2193	27	54	derived	derive	VERB
ajrhss-2193	27	55	from	from	ADP
ajrhss-2193	27	56	the	the	DET
ajrhss-2193	27	57	basic	basic	ADJ
ajrhss-2193	27	58	definition	definition	NOUN
ajrhss-2193	27	59	:	:	PUNCT
ajrhss-2193	27	60	[	[	PUNCT
ajrhss-2193	27	61	a_n	a_n	PUNCT
ajrhss-2193	27	62	=	=	PUNCT
ajrhss-2193	27	63	a	a	PRON
ajrhss-2193	27	64	+	+	X
ajrhss-2193	27	65	(	(	PUNCT
ajrhss-2193	27	66	n-1)d	n-1)d	NUM
ajrhss-2193	27	67	\	\	NOUN
ajrhss-2193	27	68	]	]	PUNCT
ajrhss-2193	27	69	where	where	SCONJ
ajrhss-2193	27	70	:	:	PUNCT
ajrhss-2193	27	71	(	(	PUNCT
ajrhss-2193	27	72	a	a	X
ajrhss-2193	27	73	)	)	PUNCT
ajrhss-2193	27	74	is	be	AUX
ajrhss-2193	27	75	the	the	DET
ajrhss-2193	27	76	first	first	ADJ
ajrhss-2193	27	77	term	term	NOUN
ajrhss-2193	27	78	,	,	PUNCT
ajrhss-2193	27	79	(	(	PUNCT
ajrhss-2193	27	80	d	d	X
ajrhss-2193	27	81	)	)	PUNCT
ajrhss-2193	27	82	is	be	AUX
ajrhss-2193	27	83	the	the	DET
ajrhss-2193	27	84	common	common	ADJ
ajrhss-2193	27	85	difference	difference	NOUN
ajrhss-2193	27	86	,	,	PUNCT
ajrhss-2193	27	87	(	(	PUNCT
ajrhss-2193	27	88	n	n	CCONJ
ajrhss-2193	27	89	)	)	PUNCT
ajrhss-2193	27	90	is	be	AUX
ajrhss-2193	27	91	the	the	DET
ajrhss-2193	27	92	term	term	NOUN
ajrhss-2193	27	93	number	number	NOUN
ajrhss-2193	27	94	.	.	PUNCT
ajrhss-2193	28	1	sum	sum	NOUN
ajrhss-2193	28	2	of	of	ADP
ajrhss-2193	28	3	the	the	DET
ajrhss-2193	28	4	first	first	ADJ
ajrhss-2193	28	5	(	(	PUNCT
ajrhss-2193	28	6	n	n	NOUN
ajrhss-2193	28	7	)	)	PUNCT
ajrhss-2193	28	8	terms	term	NOUN
ajrhss-2193	28	9	the	the	DET
ajrhss-2193	28	10	sum\	sum\	NOUN
ajrhss-2193	28	11	(	(	PUNCT
ajrhss-2193	28	12	s_n	s_n	NOUN
ajrhss-2193	28	13	\	\	NOUN
ajrhss-2193	28	14	)	)	PUNCT
ajrhss-2193	28	15	of	of	ADP
ajrhss-2193	28	16	the	the	DET
ajrhss-2193	28	17	first	first	ADJ
ajrhss-2193	28	18	\	\	PROPN
ajrhss-2193	28	19	(	(	PUNCT
ajrhss-2193	28	20	n	n	PRON
ajrhss-2193	28	21	\	\	ADJ
ajrhss-2193	28	22	)	)	PUNCT
ajrhss-2193	28	23	terms	term	NOUN
ajrhss-2193	28	24	of	of	ADP
ajrhss-2193	28	25	an	an	DET
ajrhss-2193	28	26	ap	ap	PROPN
ajrhss-2193	28	27	can	can	AUX
ajrhss-2193	28	28	be	be	AUX
ajrhss-2193	28	29	derived	derive	VERB
ajrhss-2193	28	30	as	as	SCONJ
ajrhss-2193	28	31	follows	follow	VERB
ajrhss-2193	28	32	:	:	PUNCT
ajrhss-2193	28	33	\[s_n	\[s_n	NUM
ajrhss-2193	28	34	=	=	SYM
ajrhss-2193	28	35	\sum_{i=1}^{n	\sum_{i=1}^{n	NOUN
ajrhss-2193	28	36	}	}	PUNCT
ajrhss-2193	28	37	a_i	a_i	PROPN
ajrhss-2193	29	1	=	=	PUNCT
ajrhss-2193	29	2	a	a	PROPN
ajrhss-2193	29	3	+	+	X
ajrhss-2193	29	4	(	(	PUNCT
ajrhss-2193	29	5	a	a	DET
ajrhss-2193	29	6	+	+	X
ajrhss-2193	29	7	d	d	NOUN
ajrhss-2193	29	8	)	)	PUNCT
ajrhss-2193	30	1	+	+	CCONJ
ajrhss-2193	30	2	(	(	PUNCT
ajrhss-2193	30	3	a	a	DET
ajrhss-2193	30	4	+	+	NOUN
ajrhss-2193	30	5	2d	2d	NUM
ajrhss-2193	30	6	)	)	PUNCT
ajrhss-2193	31	1	+	+	CCONJ
ajrhss-2193	31	2	\ldots	\ldots	PROPN
ajrhss-2193	31	3	+	+	NOUN
ajrhss-2193	32	1	[	[	X
ajrhss-2193	32	2	a	a	X
ajrhss-2193	32	3	+	+	X
ajrhss-2193	32	4	(	(	PUNCT
ajrhss-2193	32	5	n-1)d]\	n-1)d]\	NOUN
ajrhss-2193	32	6	]	]	X
ajrhss-2193	32	7	pairing	pair	VERB
ajrhss-2193	32	8	terms	term	NOUN
ajrhss-2193	32	9	from	from	ADP
ajrhss-2193	32	10	the	the	DET
ajrhss-2193	32	11	start	start	NOUN
ajrhss-2193	32	12	and	and	CCONJ
ajrhss-2193	32	13	end	end	VERB
ajrhss-2193	32	14	:	:	PUNCT
ajrhss-2193	32	15	\[s_n	\[s_n	PROPN
ajrhss-2193	32	16	=	=	SYM
ajrhss-2193	32	17	\left	\left	PROPN
ajrhss-2193	32	18	(	(	PUNCT
ajrhss-2193	32	19	a	a	PRON
ajrhss-2193	32	20	+	+	X
ajrhss-2193	33	1	[	[	X
ajrhss-2193	33	2	a	a	X
ajrhss-2193	33	3	+	+	X
ajrhss-2193	33	4	(	(	PUNCT
ajrhss-2193	33	5	n-1)d	n-1)d	NOUN
ajrhss-2193	33	6	]	]	X
ajrhss-2193	33	7	\right	\right	NOUN
ajrhss-2193	33	8	)	)	PUNCT
ajrhss-2193	33	9	+	+	CCONJ
ajrhss-2193	33	10	\left	\left	PROPN
ajrhss-2193	33	11	(	(	PUNCT
ajrhss-2193	33	12	[	[	X
ajrhss-2193	33	13	a	a	X
ajrhss-2193	33	14	+	+	X
ajrhss-2193	33	15	d	d	NOUN
ajrhss-2193	33	16	]	]	X
ajrhss-2193	33	17	+	+	CCONJ
ajrhss-2193	34	1	[	[	X
ajrhss-2193	34	2	a	a	X
ajrhss-2193	34	3	+	+	X
ajrhss-2193	34	4	(	(	PUNCT
ajrhss-2193	34	5	n-2)d	n-2)d	NOUN
ajrhss-2193	34	6	]	]	X
ajrhss-2193	34	7	\right	\right	NOUN
ajrhss-2193	34	8	)	)	PUNCT
ajrhss-2193	34	9	+	+	NUM
ajrhss-2193	34	10	\	\	NOUN
ajrhss-2193	34	11	...	...	PUNCT
ajrhss-2193	34	12	\	\	NOUN
ajrhss-2193	34	13	]	]	X
ajrhss-2193	34	14	each	each	DET
ajrhss-2193	34	15	pair	pair	NOUN
ajrhss-2193	34	16	sums	sum	VERB
ajrhss-2193	34	17	to	to	ADP
ajrhss-2193	34	18	:	:	PUNCT
ajrhss-2193	34	19	(	(	PUNCT
ajrhss-2193	34	20	2a	2a	NUM
ajrhss-2193	34	21	+	+	CCONJ
ajrhss-2193	34	22	(	(	PUNCT
ajrhss-2193	34	23	n-1)d	n-1)d	NOUN
ajrhss-2193	34	24	)	)	PUNCT
ajrhss-2193	34	25	there	there	PRON
ajrhss-2193	34	26	are	be	VERB
ajrhss-2193	34	27	\	\	NOUN
ajrhss-2193	34	28	(	(	PUNCT
ajrhss-2193	34	29	n/2	n/2	PROPN
ajrhss-2193	34	30	\	\	NOUN
ajrhss-2193	34	31	)	)	PUNCT
ajrhss-2193	35	1	such	such	ADJ
ajrhss-2193	35	2	pairs	pair	NOUN
ajrhss-2193	35	3	(	(	PUNCT
ajrhss-2193	35	4	if	if	SCONJ
ajrhss-2193	35	5	\	\	PROPN
ajrhss-2193	35	6	(	(	PUNCT
ajrhss-2193	35	7	n	n	PRON
ajrhss-2193	35	8	\	\	NOUN
ajrhss-2193	35	9	)	)	PUNCT
ajrhss-2193	35	10	is	be	AUX
ajrhss-2193	35	11	even	even	ADV
ajrhss-2193	35	12	):	):	PUNCT
ajrhss-2193	35	13	s_n	s_n	SYM
ajrhss-2193	35	14	=	=	SYM
ajrhss-2193	35	15	\frac{n}{2	\frac{n}{2	PROPN
ajrhss-2193	35	16	}	}	PUNCT
ajrhss-2193	35	17	(	(	PUNCT
ajrhss-2193	35	18	2a	2a	NUM
ajrhss-2193	35	19	+	+	CCONJ
ajrhss-2193	35	20	(	(	PUNCT
ajrhss-2193	35	21	n-1)d	n-1)d	PROPN
ajrhss-2193	35	22	)	)	PUNCT
ajrhss-2193	35	23	american	american	ADJ
ajrhss-2193	35	24	journal	journal	PROPN
ajrhss-2193	35	25	of	of	ADP
ajrhss-2193	35	26	research	research	NOUN
ajrhss-2193	35	27	in	in	ADP
ajrhss-2193	35	28	humanities	humanity	NOUN
ajrhss-2193	35	29	and	and	CCONJ
ajrhss-2193	35	30	social	social	ADJ
ajrhss-2193	35	31	sciences	science	NOUN
ajrhss-2193	35	32	volume	volume	NOUN
ajrhss-2193	35	33	25	25	NUM
ajrhss-2193	35	34	june	june	PROPN
ajrhss-2193	35	35	2024	2024	NUM
ajrhss-2193	35	36	p	p	NOUN
ajrhss-2193	35	37	a	a	DET
ajrhss-2193	35	38	g	g	NOUN
ajrhss-2193	35	39	e	e	NOUN
ajrhss-2193	35	40	|	|	ADV
ajrhss-2193	35	41	40	40	NUM
ajrhss-2193	35	42	www.americanjournal.org	www.americanjournal.org	NOUN
ajrhss-2193	35	43	alternatively	alternatively	ADV
ajrhss-2193	35	44	,	,	PUNCT
ajrhss-2193	35	45	using	use	VERB
ajrhss-2193	35	46	the	the	DET
ajrhss-2193	35	47	last	last	ADJ
ajrhss-2193	35	48	term	term	NOUN
ajrhss-2193	35	49	\	\	PROPN
ajrhss-2193	35	50	(	(	PUNCT
ajrhss-2193	35	51	l	l	NOUN
ajrhss-2193	35	52	\	\	PROPN
ajrhss-2193	35	53	):	):	PUNCT
ajrhss-2193	35	54	s_n	s_n	PUNCT
ajrhss-2193	35	55	=	=	SYM
ajrhss-2193	35	56	\frac{n}{2	\frac{n}{2	PROPN
ajrhss-2193	35	57	}	}	PUNCT
ajrhss-2193	35	58	(	(	PUNCT
ajrhss-2193	35	59	a	a	DET
ajrhss-2193	35	60	+	+	NUM
ajrhss-2193	35	61	l	l	NOUN
ajrhss-2193	35	62	)	)	PUNCT
ajrhss-2193	35	63	applications	application	NOUN
ajrhss-2193	35	64	in	in	ADP
ajrhss-2193	35	65	real	real	ADJ
ajrhss-2193	35	66	life	life	NOUN
ajrhss-2193	35	67	1	1	NUM
ajrhss-2193	35	68	.	.	PUNCT
ajrhss-2193	35	69	finance	finance	NOUN
ajrhss-2193	35	70	and	and	CCONJ
ajrhss-2193	35	71	economics	economic	NOUN
ajrhss-2193	35	72	:	:	PUNCT
ajrhss-2193	35	73	calculation	calculation	NOUN
ajrhss-2193	35	74	of	of	ADP
ajrhss-2193	35	75	loan	loan	NOUN
ajrhss-2193	35	76	repayments	repayment	NOUN
ajrhss-2193	35	77	.	.	PUNCT
ajrhss-2193	36	1	finding	find	VERB
ajrhss-2193	36	2	total	total	ADJ
ajrhss-2193	36	3	interest	interest	NOUN
ajrhss-2193	36	4	earned	earn	VERB
ajrhss-2193	36	5	over	over	ADP
ajrhss-2193	36	6	a	a	DET
ajrhss-2193	36	7	period	period	NOUN
ajrhss-2193	36	8	with	with	ADP
ajrhss-2193	36	9	regular	regular	ADJ
ajrhss-2193	36	10	deposits	deposit	NOUN
ajrhss-2193	36	11	.	.	PUNCT
ajrhss-2193	37	1	2	2	X
ajrhss-2193	37	2	.	.	X
ajrhss-2193	37	3	physics	physic	NOUN
ajrhss-2193	37	4	:	:	PUNCT
ajrhss-2193	37	5	analysis	analysis	NOUN
ajrhss-2193	37	6	of	of	ADP
ajrhss-2193	37	7	uniform	uniform	ADJ
ajrhss-2193	37	8	motion	motion	NOUN
ajrhss-2193	37	9	where	where	SCONJ
ajrhss-2193	37	10	distance	distance	NOUN
ajrhss-2193	37	11	increases	increase	VERB
ajrhss-2193	37	12	linearly	linearly	ADV
ajrhss-2193	37	13	over	over	ADP
ajrhss-2193	37	14	time	time	NOUN
ajrhss-2193	37	15	.	.	PUNCT
ajrhss-2193	38	1	3	3	X
ajrhss-2193	38	2	.	.	X
ajrhss-2193	38	3	engineering	engineering	NOUN
ajrhss-2193	38	4	:	:	PUNCT
ajrhss-2193	38	5	stress	stress	NOUN
ajrhss-2193	38	6	distribution	distribution	NOUN
ajrhss-2193	38	7	in	in	ADP
ajrhss-2193	38	8	materials	material	NOUN
ajrhss-2193	38	9	can	can	AUX
ajrhss-2193	38	10	sometimes	sometimes	ADV
ajrhss-2193	38	11	follow	follow	VERB
ajrhss-2193	38	12	an	an	DET
ajrhss-2193	38	13	arithmetic	arithmetic	ADJ
ajrhss-2193	38	14	sequence	sequence	NOUN
ajrhss-2193	38	15	.	.	PUNCT
ajrhss-2193	39	1	signal	signal	NOUN
ajrhss-2193	39	2	processing	processing	NOUN
ajrhss-2193	39	3	and	and	CCONJ
ajrhss-2193	39	4	sampling	sampling	NOUN
ajrhss-2193	39	5	.	.	PUNCT
ajrhss-2193	40	1	4	4	X
ajrhss-2193	40	2	.	.	X
ajrhss-2193	40	3	computer	computer	NOUN
ajrhss-2193	40	4	science	science	NOUN
ajrhss-2193	40	5	:	:	PUNCT
ajrhss-2193	40	6	used	use	VERB
ajrhss-2193	40	7	in	in	ADP
ajrhss-2193	40	8	algorithms	algorithm	NOUN
ajrhss-2193	40	9	and	and	CCONJ
ajrhss-2193	40	10	coding	code	VERB
ajrhss-2193	40	11	challenges	challenge	NOUN
ajrhss-2193	40	12	involving	involve	VERB
ajrhss-2193	40	13	sequences	sequence	NOUN
ajrhss-2193	40	14	.	.	PUNCT
ajrhss-2193	41	1	analysis	analysis	NOUN
ajrhss-2193	41	2	of	of	ADP
ajrhss-2193	41	3	data	datum	NOUN
ajrhss-2193	41	4	structures	structure	NOUN
ajrhss-2193	41	5	like	like	ADP
ajrhss-2193	41	6	arrays	array	NOUN
ajrhss-2193	41	7	.	.	PUNCT
ajrhss-2193	42	1	5	5	X
ajrhss-2193	42	2	.	.	X
ajrhss-2193	42	3	daily	daily	ADJ
ajrhss-2193	42	4	life	life	NOUN
ajrhss-2193	42	5	:	:	PUNCT
ajrhss-2193	42	6	planning	planning	NOUN
ajrhss-2193	42	7	and	and	CCONJ
ajrhss-2193	42	8	scheduling	schedule	VERB
ajrhss-2193	42	9	repetitive	repetitive	ADJ
ajrhss-2193	42	10	tasks	task	NOUN
ajrhss-2193	42	11	.	.	PUNCT
ajrhss-2193	43	1	managing	manage	VERB
ajrhss-2193	43	2	and	and	CCONJ
ajrhss-2193	43	3	predicting	predict	VERB
ajrhss-2193	43	4	costs	cost	NOUN
ajrhss-2193	43	5	,	,	PUNCT
ajrhss-2193	43	6	like	like	ADP
ajrhss-2193	43	7	subscription	subscription	NOUN
ajrhss-2193	43	8	services	service	NOUN
ajrhss-2193	43	9	with	with	ADP
ajrhss-2193	43	10	regular	regular	ADJ
ajrhss-2193	43	11	price	price	NOUN
ajrhss-2193	43	12	increases	increase	NOUN
ajrhss-2193	43	13	.	.	PUNCT
ajrhss-2193	44	1	advanced	advanced	ADJ
ajrhss-2193	44	2	concepts	concept	NOUN
ajrhss-2193	44	3	arithmetic	arithmetic	ADJ
ajrhss-2193	44	4	-	-	PUNCT
ajrhss-2193	44	5	geometric	geometric	ADJ
ajrhss-2193	44	6	progression	progression	NOUN
ajrhss-2193	44	7	(	(	PUNCT
ajrhss-2193	44	8	agp	agp	NOUN
ajrhss-2193	44	9	)	)	PUNCT
ajrhss-2193	44	10	an	an	DET
ajrhss-2193	44	11	agp	agp	NOUN
ajrhss-2193	44	12	is	be	AUX
ajrhss-2193	44	13	a	a	DET
ajrhss-2193	44	14	sequence	sequence	NOUN
ajrhss-2193	44	15	in	in	ADP
ajrhss-2193	44	16	which	which	PRON
ajrhss-2193	44	17	each	each	DET
ajrhss-2193	44	18	term	term	NOUN
ajrhss-2193	44	19	is	be	AUX
ajrhss-2193	44	20	the	the	DET
ajrhss-2193	44	21	product	product	NOUN
ajrhss-2193	44	22	of	of	ADP
ajrhss-2193	44	23	the	the	DET
ajrhss-2193	44	24	corresponding	corresponding	ADJ
ajrhss-2193	44	25	terms	term	NOUN
ajrhss-2193	44	26	of	of	ADP
ajrhss-2193	44	27	an	an	DET
ajrhss-2193	44	28	arithmetic	arithmetic	ADJ
ajrhss-2193	44	29	progression	progression	NOUN
ajrhss-2193	44	30	and	and	CCONJ
ajrhss-2193	44	31	a	a	DET
ajrhss-2193	44	32	geometric	geometric	ADJ
ajrhss-2193	44	33	progression	progression	NOUN
ajrhss-2193	44	34	.	.	PUNCT
ajrhss-2193	45	1	for	for	ADP
ajrhss-2193	45	2	instance	instance	NOUN
ajrhss-2193	45	3	,	,	PUNCT
ajrhss-2193	45	4	if	if	SCONJ
ajrhss-2193	45	5	\	\	PROPN
ajrhss-2193	45	6	(	(	PUNCT
ajrhss-2193	45	7	a	a	PRON
ajrhss-2193	45	8	,	,	PUNCT
ajrhss-2193	45	9	a+d	a+d	NUM
ajrhss-2193	45	10	,	,	PUNCT
ajrhss-2193	45	11	a+2d	a+2d	PROPN
ajrhss-2193	45	12	,	,	PUNCT
ajrhss-2193	45	13	\	\	PROPN
ajrhss-2193	45	14	...	...	PUNCT
ajrhss-2193	45	15	\	\	X
ajrhss-2193	45	16	)	)	PUNCT
ajrhss-2193	45	17	is	be	AUX
ajrhss-2193	45	18	an	an	DET
ajrhss-2193	45	19	ap	ap	PROPN
ajrhss-2193	45	20	and	and	CCONJ
ajrhss-2193	45	21	\	\	PROPN
ajrhss-2193	45	22	(	(	PUNCT
ajrhss-2193	45	23	b	b	PROPN
ajrhss-2193	45	24	,	,	PUNCT
ajrhss-2193	45	25	br	br	PROPN
ajrhss-2193	45	26	,	,	PUNCT
ajrhss-2193	45	27	br^2	br^2	PROPN
ajrhss-2193	45	28	,	,	PUNCT
ajrhss-2193	45	29	\	\	PROPN
ajrhss-2193	45	30	...	...	PUNCT
ajrhss-2193	46	1	\	\	NOUN
ajrhss-2193	46	2	)	)	PUNCT
ajrhss-2193	46	3	is	be	AUX
ajrhss-2193	46	4	a	a	DET
ajrhss-2193	46	5	gp	gp	NOUN
ajrhss-2193	46	6	,	,	PUNCT
ajrhss-2193	46	7	then	then	ADV
ajrhss-2193	46	8	\	\	PROPN
ajrhss-2193	46	9	(	(	PUNCT
ajrhss-2193	46	10	ab	ab	PROPN
ajrhss-2193	46	11	,	,	PUNCT
ajrhss-2193	46	12	(	(	PUNCT
ajrhss-2193	46	13	a+d)br	a+d)br	X
ajrhss-2193	46	14	,	,	PUNCT
ajrhss-2193	46	15	(	(	PUNCT
ajrhss-2193	46	16	a+2d)br^2	a+2d)br^2	PROPN
ajrhss-2193	46	17	,	,	PUNCT
ajrhss-2193	46	18	\	\	PROPN
ajrhss-2193	46	19	...	...	PUNCT
ajrhss-2193	47	1	\	\	NOUN
ajrhss-2193	47	2	)	)	PUNCT
ajrhss-2193	47	3	is	be	AUX
ajrhss-2193	47	4	an	an	DET
ajrhss-2193	47	5	agp	agp	NOUN
ajrhss-2193	47	6	.	.	PUNCT
ajrhss-2193	48	1	solving	solve	VERB
ajrhss-2193	48	2	problems	problem	NOUN
ajrhss-2193	48	3	involving	involve	VERB
ajrhss-2193	48	4	ap	ap	PROPN
ajrhss-2193	48	5	finding	find	VERB
ajrhss-2193	48	6	missing	miss	VERB
ajrhss-2193	48	7	terms	term	NOUN
ajrhss-2193	48	8	:	:	PUNCT
ajrhss-2193	48	9	given	give	VERB
ajrhss-2193	48	10	a	a	DET
ajrhss-2193	48	11	few	few	ADJ
ajrhss-2193	48	12	terms	term	NOUN
ajrhss-2193	48	13	in	in	ADP
ajrhss-2193	48	14	an	an	DET
ajrhss-2193	48	15	ap	ap	PROPN
ajrhss-2193	48	16	,	,	PUNCT
ajrhss-2193	48	17	you	you	PRON
ajrhss-2193	48	18	can	can	AUX
ajrhss-2193	48	19	find	find	VERB
ajrhss-2193	48	20	missing	miss	VERB
ajrhss-2193	48	21	terms	term	NOUN
ajrhss-2193	48	22	by	by	ADP
ajrhss-2193	48	23	solving	solve	VERB
ajrhss-2193	48	24	linear	linear	ADJ
ajrhss-2193	48	25	equations	equation	NOUN
ajrhss-2193	48	26	.	.	PUNCT
ajrhss-2193	49	1	transformations	transformation	NOUN
ajrhss-2193	49	2	:	:	PUNCT
ajrhss-2193	49	3	understanding	understand	VERB
ajrhss-2193	49	4	shifts	shift	NOUN
ajrhss-2193	49	5	and	and	CCONJ
ajrhss-2193	49	6	transformations	transformation	NOUN
ajrhss-2193	49	7	of	of	ADP
ajrhss-2193	49	8	sequences	sequence	NOUN
ajrhss-2193	49	9	to	to	PART
ajrhss-2193	49	10	solve	solve	VERB
ajrhss-2193	49	11	complex	complex	ADJ
ajrhss-2193	49	12	problems	problem	NOUN
ajrhss-2193	49	13	.	.	PUNCT
ajrhss-2193	50	1	example	example	NOUN
ajrhss-2193	50	2	problem	problem	NOUN
ajrhss-2193	50	3	problem	problem	NOUN
ajrhss-2193	50	4	:	:	PUNCT
ajrhss-2193	50	5	find	find	VERB
ajrhss-2193	50	6	the	the	DET
ajrhss-2193	50	7	10th	10th	ADJ
ajrhss-2193	50	8	term	term	NOUN
ajrhss-2193	50	9	and	and	CCONJ
ajrhss-2193	50	10	the	the	DET
ajrhss-2193	50	11	sum	sum	NOUN
ajrhss-2193	50	12	of	of	ADP
ajrhss-2193	50	13	the	the	DET
ajrhss-2193	50	14	first	first	ADJ
ajrhss-2193	50	15	10	10	NUM
ajrhss-2193	50	16	terms	term	NOUN
ajrhss-2193	50	17	of	of	ADP
ajrhss-2193	50	18	the	the	DET
ajrhss-2193	50	19	ap	ap	PROPN
ajrhss-2193	50	20	where	where	SCONJ
ajrhss-2193	50	21	the	the	DET
ajrhss-2193	50	22	first	first	ADJ
ajrhss-2193	50	23	term	term	NOUN
ajrhss-2193	50	24	is	be	AUX
ajrhss-2193	50	25	2	2	NUM
ajrhss-2193	50	26	and	and	CCONJ
ajrhss-2193	50	27	the	the	DET
ajrhss-2193	50	28	common	common	ADJ
ajrhss-2193	50	29	difference	difference	NOUN
ajrhss-2193	50	30	is	be	AUX
ajrhss-2193	50	31	5	5	NUM
ajrhss-2193	50	32	.	.	PUNCT
ajrhss-2193	51	1	solution	solution	NOUN
ajrhss-2193	51	2	:	:	PUNCT
ajrhss-2193	51	3	1	1	X
ajrhss-2193	51	4	.	.	NUM
ajrhss-2193	51	5	10th	10th	ADJ
ajrhss-2193	51	6	term	term	NOUN
ajrhss-2193	51	7	:	:	PUNCT
ajrhss-2193	51	8	\[a_{10	\[a_{10	PROPN
ajrhss-2193	51	9	}	}	PUNCT
ajrhss-2193	51	10	=	=	SYM
ajrhss-2193	51	11	2	2	NUM
ajrhss-2193	51	12	+	+	CCONJ
ajrhss-2193	51	13	(	(	PUNCT
ajrhss-2193	51	14	10	10	NUM
ajrhss-2193	51	15	-	-	SYM
ajrhss-2193	51	16	1	1	NUM
ajrhss-2193	51	17	)	)	PUNCT
ajrhss-2193	51	18	\cdot	\cdot	NOUN
ajrhss-2193	51	19	5	5	NUM
ajrhss-2193	51	20	=	=	SYM
ajrhss-2193	51	21	2	2	NUM
ajrhss-2193	51	22	+	+	NUM
ajrhss-2193	51	23	45	45	NUM
ajrhss-2193	51	24	=	=	SYM
ajrhss-2193	51	25	47	47	NUM
ajrhss-2193	51	26	\	\	NOUN
ajrhss-2193	51	27	]	]	X
ajrhss-2193	51	28	2	2	X
ajrhss-2193	51	29	.	.	X
ajrhss-2193	51	30	sum	sum	NOUN
ajrhss-2193	51	31	of	of	ADP
ajrhss-2193	51	32	the	the	DET
ajrhss-2193	51	33	first	first	ADJ
ajrhss-2193	51	34	10	10	NUM
ajrhss-2193	51	35	terms	term	NOUN
ajrhss-2193	51	36	:	:	PUNCT
ajrhss-2193	51	37	\[s_{10	\[s_{10	NOUN
ajrhss-2193	51	38	}	}	PUNCT
ajrhss-2193	51	39	=	=	SYM
ajrhss-2193	51	40	\frac{10}{2	\frac{10}{2	ADJ
ajrhss-2193	51	41	}	}	PUNCT
ajrhss-2193	51	42	(	(	PUNCT
ajrhss-2193	51	43	2	2	NUM
ajrhss-2193	51	44	\cdot	\cdot	NOUN
ajrhss-2193	51	45	2	2	NUM
ajrhss-2193	51	46	+	+	CCONJ
ajrhss-2193	51	47	(	(	PUNCT
ajrhss-2193	51	48	10	10	NUM
ajrhss-2193	51	49	-	-	SYM
ajrhss-2193	51	50	1	1	NUM
ajrhss-2193	51	51	)	)	PUNCT
ajrhss-2193	51	52	\cdot	\cdot	NOUN
ajrhss-2193	51	53	5	5	NUM
ajrhss-2193	51	54	)	)	PUNCT
ajrhss-2193	51	55	=	=	SYM
ajrhss-2193	51	56	5	5	NUM
ajrhss-2193	51	57	\cdot	\cdot	NOUN
ajrhss-2193	51	58	(	(	PUNCT
ajrhss-2193	51	59	4	4	NUM
ajrhss-2193	51	60	+	+	SYM
ajrhss-2193	51	61	45	45	NUM
ajrhss-2193	51	62	)	)	PUNCT
ajrhss-2193	51	63	=	=	NOUN
ajrhss-2193	51	64	5	5	NUM
ajrhss-2193	51	65	\cdot	\cdot	NOUN
ajrhss-2193	51	66	49	49	NUM
ajrhss-2193	51	67	=	=	SYM
ajrhss-2193	51	68	245\	245\	NOUN
ajrhss-2193	51	69	]	]	X
ajrhss-2193	51	70	these	these	DET
ajrhss-2193	51	71	calculations	calculation	NOUN
ajrhss-2193	51	72	illustrate	illustrate	VERB
ajrhss-2193	51	73	the	the	DET
ajrhss-2193	51	74	practical	practical	ADJ
ajrhss-2193	51	75	use	use	NOUN
ajrhss-2193	51	76	of	of	ADP
ajrhss-2193	51	77	the	the	DET
ajrhss-2193	51	78	ap	ap	PROPN
ajrhss-2193	51	79	formulas	formula	NOUN
ajrhss-2193	51	80	.	.	PUNCT
ajrhss-2193	52	1	references	reference	NOUN
ajrhss-2193	52	2	1	1	NUM
ajrhss-2193	52	3	.	.	PUNCT
ajrhss-2193	53	1	"	"	PUNCT
ajrhss-2193	53	2	elementary	elementary	ADJ
ajrhss-2193	53	3	number	number	NOUN
ajrhss-2193	53	4	theory	theory	NOUN
ajrhss-2193	53	5	"	"	PUNCT
ajrhss-2193	53	6	by	by	ADP
ajrhss-2193	53	7	david	david	PROPN
ajrhss-2193	53	8	m.	m.	PROPN
ajrhss-2193	53	9	burton	burton	PROPN
ajrhss-2193	53	10	2	2	NUM
ajrhss-2193	53	11	.	.	PUNCT
ajrhss-2193	54	1	"	"	PUNCT
ajrhss-2193	54	2	introduction	introduction	NOUN
ajrhss-2193	54	3	to	to	ADP
ajrhss-2193	54	4	the	the	DET
ajrhss-2193	54	5	theory	theory	NOUN
ajrhss-2193	54	6	of	of	ADP
ajrhss-2193	54	7	numbers	number	NOUN
ajrhss-2193	54	8	"	"	PUNCT
ajrhss-2193	54	9	by	by	ADP
ajrhss-2193	54	10	g.h	g.h	PROPN
ajrhss-2193	54	11	.	.	PROPN
ajrhss-2193	54	12	hardy	hardy	PROPN
ajrhss-2193	54	13	and	and	CCONJ
ajrhss-2193	54	14	e.m	e.m	PROPN
ajrhss-2193	54	15	.	.	PROPN
ajrhss-2193	54	16	wright	wright	PROPN
ajrhss-2193	54	17	:	:	PUNCT
ajrhss-2193	55	1	3	3	X
ajrhss-2193	55	2	.	.	X
ajrhss-2193	55	3	"	"	PUNCT
ajrhss-2193	55	4	discrete	discrete	ADJ
ajrhss-2193	55	5	mathematics	mathematic	NOUN
ajrhss-2193	55	6	and	and	CCONJ
ajrhss-2193	55	7	its	its	PRON
ajrhss-2193	55	8	applications	application	NOUN
ajrhss-2193	55	9	"	"	PUNCT
ajrhss-2193	55	10	by	by	ADP
ajrhss-2193	55	11	kenneth	kenneth	PROPN
ajrhss-2193	55	12	h.	h.	PROPN
ajrhss-2193	55	13	rosen	rosen	PROPN
ajrhss-2193	55	14	4	4	X
ajrhss-2193	55	15	.	.	PUNCT
ajrhss-2193	56	1	"	"	PUNCT
ajrhss-2193	56	2	mathematics	mathematic	NOUN
ajrhss-2193	56	3	:	:	PUNCT
ajrhss-2193	56	4	its	its	PRON
ajrhss-2193	56	5	content	content	NOUN
ajrhss-2193	56	6	,	,	PUNCT
ajrhss-2193	56	7	methods	method	NOUN
ajrhss-2193	56	8	and	and	CCONJ
ajrhss-2193	56	9	meaning	meaning	NOUN
ajrhss-2193	56	10	"	"	PUNCT
ajrhss-2193	56	11	by	by	ADP
ajrhss-2193	56	12	a.d	a.d	PROPN
ajrhss-2193	56	13	.	.	PROPN
ajrhss-2193	56	14	aleksandrov	aleksandrov	PROPN
ajrhss-2193	56	15	,	,	PUNCT
ajrhss-2193	56	16	a.n	a.n	PROPN
ajrhss-2193	56	17	.	.	PROPN
ajrhss-2193	56	18	kolmogorov	kolmogorov	PROPN
ajrhss-2193	56	19	,	,	PUNCT
ajrhss-2193	56	20	and	and	CCONJ
ajrhss-2193	56	21	m.a	m.a	PROPN
ajrhss-2193	56	22	.	.	PROPN
ajrhss-2193	56	23	lavrent'ev	lavrent'ev	PROPN
ajrhss-2193	56	24	5	5	NUM
ajrhss-2193	56	25	.	.	PUNCT
ajrhss-2193	56	26	"	"	PUNCT
ajrhss-2193	56	27	principles	principle	NOUN
ajrhss-2193	56	28	of	of	ADP
ajrhss-2193	56	29	mathematical	mathematical	ADJ
ajrhss-2193	56	30	analysis	analysis	NOUN
ajrhss-2193	56	31	"	"	PUNCT
ajrhss-2193	56	32	by	by	ADP
ajrhss-2193	56	33	walter	walter	PROPN
ajrhss-2193	56	34	rudin	rudin	PROPN
ajrhss-2193	56	35	.	.	PUNCT