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-

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), \dots \dots \dots \dots

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, \dots

Answered by Tinkutara last updated on 14/Mar/18

L e t S_{n} = 1 + 2 + 3 + \dots + (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!