I have a question from my discrete mathematics class. The prompt is as follows:
$\textit{Using the idea of a telescoping series, find a closed formula for $a_{k}$ if …}$
$\sum_{k = 1}^{n} a_{k} = 3n^{2} + 5n$
I don't understand how to solve this problem. I though the idea of a telescoping series was that if you write out the whole sum from $k = 1$ to $n$, the inner pieces cancel each other out. That doesn't work if we're adding $5n$ instead of subtracting something.
Any advice on how to proceed would be greatly appreciated. Thank you.
Best Answer
Hint:
Show with telescoping sums that
$$a_n-a_1=\sum_{k=2}^n(a_k-a_{k-1})$$
And recall that
$$\sum(x+y)=\left(\sum x\right)+\left(\sum y\right)$$