# In the sequence 1,2,2,4,4,4,4,8,8,8,8,8,8,8,8 where n consective terms have the value n, then 1025th term is?

Jun 24, 2018

$1024$

#### Explanation:

This sequence is composed by powers of $2$, repeated ${2}^{n}$ times.

In fact, you have ${2}^{0} = 1$, and $1$ appears one time.

Then, ${2}^{1} = 2$, and $2$ appears twice.

Then, ${2}^{2} = 4$ and $4$ appears four times.

Then, ${2}^{3} = 8$, and $8$ appears eight times.

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

Also, note that the first $1$ is in first position, the first $2$ is in second position, the first $4$ is in fourth position, and so on. This makes it easy to write the sequence without repetition, like this:

• 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 five hundred and twelve $512$s, starting from position $512$
• There are one thousand and twenty-four $1024$s, starting from position $1024$

So, the ${1025}^{\text{th}}$ element is $1024$.

In general, the ${k}^{t h}$ term of this sequence is ${2}^{n}$, if ${2}^{n} \setminus \le k < {2}^{n + 1}$