Question

Onsider the arithmetic series: 1 + 9 + 17 + 25 + ... write a formula for the sum of the first n terms in this series.

Answer 1

We can see this is a summation of multiples of 8 plus 1, which starts at 1.

This can be written in the following way:
$\Sigma_{m=0}^{n}(8m+1)$

Lets verify the summation expression by doing the sequence for the first 4 elements (hence $n=3$, as it starts from 0):
$\Sigma_{m=0}^{3}(8m+1)=[8(0)+1]+[8(1)+1]+[8(2)+1]+[8(3)+1]=$
$1+9+17+25$

As you can see, the generalization is correct.