Question

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?

Answer 1

`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^{"th"}` element is `1024`.

In general, the `k^{th}` term of this sequence is `2^n`, if `2^n \le k < 2^{n+1}`