Question

In 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, the 1025th term is

• A. $2^9$
• B. $2^{10}$
• C. $2^{11}$
• D. $2^8$

Answer 1

Answer: (B)

$2^{10}$

Let $m=$ Number of times a number $n$ is repeated in the given sequence $\left\{n\right\}$
$\therefore m=1,2,4,\mathrm{8...}=2^0,2^1,2^2,2^3...$
Thus, sequence $\left\{m\right\}$ is a GP
$N^{th}$ term of GP $=2^{N-1}$ where $N=\{1,2,3,...\}$
Sum of N terms, $S_N=2^N-1$
The ${1025}^{th}$ term of sequence $\left\{n\right\}$ will be minimum $m$ such that $S_N\ge 1025$
Thus, $2^N-1\ge 1025$
$\Rightarrow 2^N\ge 1026$
$\therefore$ Minimum m for which $2^N-1\ge 1025$ is $2048=2^{11}$ which is the ${12}^{th}$ term
$\therefore {1025}^{th}$ term of $\left\{n\right\}$ is ${11}^{th}$ term of $\left\{m\right\}=2^{10}$
Thus ${1025}^{th}$ term of original sequence is $2^{10}$