The question ask us to guess an explicit formula for the sequence

$$s_k = s_{k-1} + 2k ,$$ for all integers $k$ greater than or equal to one and

$s_0 = 3$

Can someone help me with this? Because I don't really understand how to do this.

Any help will be appreciated.

## Best Answer

Hint: Moving the $s_{k-1}$ term to the LHS of the equation, the recurrence relation reads:

$$s_{k}-s_{k-1}=2k.$$

You can solve this recurrence relation by simply summing both sides over $k$:

$$\sum_{k=1}^{\infty}(s_{k}-s_{k-1})=2\sum_{k=1}^{\infty}k.$$

The sum on the LHS telescopes, while the sum on the RHS is a straightforward arithmetic progression. There are standard methods for finding nice closed forms for both series.