How do you write a sum in expanded form?

Perhaps you just mean to convert it from “summation form” (“sigma form”) to a written out form?
For something like n∑i=1i2, the summation symbol Sigma just means to “add up”. Putting an i=1 underneath the summation symbol means to start the value of i at 1. It is then assumed that i keeps increasing by 1 until it reaches i=n, where n is the number above the summation symbol. The i^2 represents the formula for the terms that get added, first when i=1, then i=2, then i=3, etc…, until i=n.
Therefore, the answer would be sum_{i=1}^{n}i^{2}=1^{2}+2^{2}+3^{2}+cdots+(n-1)^2+n^2.
This example is interesting in that there is a shortcut formula for adding up the first n squares: it equals
frac{n(n+1)(2n+1)}{6}=frac{1}{3}n^[3}+frac{1}{2}n^{2}+frac{1}{6}n.
You should take the time to check that this works when, for instance, n=5.

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