Disclaimer: This is for homework, but I'm just stuck on this one small part of a larger problem.

I'm having trouble figuring out how to get the following summation in closed form.

$$\sum_{j=1}^i 4ij$$

Since the index is j, am I able to move the i out of the summation as a constant the way I am with the 4? If I do that, do I need to square the i because it is the stopping point of the summation?

Am I able to separate the summation into two, like this?

$$\left(\sum_{j=1}^i 4i\right)\left(\sum_{j=1}^i j\right)$$


The $i$ is indeed a constant, so you treat it exactly as like the $4$:


It might help you to write out some terms: the sum is

$$4i\cdot1+4i\cdot2+4i\cdot3+\ldots+4i\cdot i\;,$$

which clearly has a factor of $4i$ in each term that can be factored out to yield



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