Question

Find number of digits ${1025}^{th}$ terms of the sequence $1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,........$ ?

Answer 1

The given sequence is $1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,....$

The sequence is composed by powers of $2$ and repeated $2^n$ times.

We have,

$2^0=1$ and $1$ appears one time.

$2^1=2$ and $2$ appears twice.

$2^2=4$ and $4$ appears four times.

$2^3=8$ and $8$ appears eight times.

So, the next number will be the next power of $2$ and it will appear $2^n$ times.

also, the first $1$ is in first position, the first $2$ is in second position, the first $4$ is in fourth position and so on.

So,

there is one $1$, starting from position $1$

there are two $2$s, starting from position $2$

there are four $4$s, starting from position $4$

there are eight $8$s, starting from position $8$

$....$

there are one thousand and twenty four $1024$s, starting from position $1024$

So, the ${1025}^{th}$ number is $1024.$