Question

What is the formula for the general term of the sequence `13, 15, 19, 27, 43` ?

Answer 1

`a_n = 2^n+11`

Explanation:

Given:

`13, 15, 19, 27, 43`

The differences between pairs of successive terms are:

`2, 4, 8, 16`

These are powers of `2` as observed in the question.

Note that if we take the differences again then we get the sequence:

`2, 4, 8`

Notice that we would get this sequence of powers of `2` if every term of the original sequence was offset by the same constant. That is, if our original sequence was:

`2+c, 4+c, 8+c, 16+c, 32+c`

then the differences would be:

`2, 4, 8, 16`

In fact we find that this is the case, with `c=11` and hence we can write a formula for the general term:

`a_n = 2^n+11`