Let the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ... where n consecutive terms have the value n, then 1025th term is

Question

Let the sequence 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ... where n consecutive terms have the value n, then 1025th term is (A) 29 (B) 210 (C) 211 (D) 28

Tardigrade Admin · Accepted Answer

Let the 1025 th term = 2n Then 1+2+4+8+.....+2n-1 < 1025 le 1+2+4+8+.....+2n ∴ 2n-1 <1025 <2n+1 -1 or 2n < 1026<2n+1 ⇒ n= 10 ∴ reqd. term = 210

Q. Let the sequence $1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, ...$ where n consecutive terms have the value n, then 1025th term is

$2^{9}$

$2^{10}$

$2^{11}$

$2^{8}$

Let the $1025 t h$ term $= 2^{n}$
Then $1 + 2 + 4 + 8 + ..... + 2^{n - 1} < 1025$
$\leq 1 + 2 + 4 + 8 + ..... + 2^{n} < b r / > ∴ 2^{n} - 1 < 1025 < 2^{n + 1} - 1$
or $2^{n} < 1026 < 2^{n + 1}$
$\Rightarrow n = 10$
$∴$ reqd. term $= 2^{10}$

Q. Let the sequence 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,... where n consecutive terms have the value n, then 1025th term is

29

210

211

28