Consider a sequence in the form of groups (1),(2,2),(3,3,3),(4,4,4,4), (5,5,5,5,5),............ then the 2000th term of the above sequence is : ?


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Question Number 31771 by rahul 19 last updated on 14/Mar/18

$C o n s i d e r a s e q u e n c e i n t h e f o r m o f$ $g r o u p s (1), (2, 2), (3, 3, 3), (4, 4, 4, 4),$ $(5, 5, 5, 5, 5), . . . . . . . . . . . .$ $t h e n t h e 2000 t h t e r m o f t h e a b o v e$ $s e q u e n c e i s : ?$
Commented by Joel578 last updated on 14/Mar/18

$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, . . .$
	Answered by Tinkutara last updated on 14/Mar/18

	$L e t S_{n} = 1 + 2 + 3 + . . . + (n t e r m s)$ $S_{n} = \frac{n (n + 1)}{2}$ $N o w o b s e r v e S_{4} = 10 . I t m e a n s 10^{t h} t e r m$ $i s t h e l a s t ‘ ‘ 4^{″} .$ $S i m i l a r l y S_{5} = 15 m e a n s 15^{t h} t e r m$ $i s t h e l a s t ‘ ‘ 5^{″} .$ $⋮$ $\frac{n (n + 1)}{2} = 2000$ $n \approx 62.75$ $S_{62} = 1953 \Rightarrow 1953^{r d} t e r m i s t h e l a s t 62 .$ $S_{63} = 2016 \Rightarrow 2016^{t h} t e r m i s t h e l a s t 63 .$ $I n b e t w e e n, 2000^{t h} t e r m i s 63 .$
	Commented by rahul 19 last updated on 14/Mar/18

	$W O W!$
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