[Math] Guess explicit formula using iteration


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:


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


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.

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