I am trying to understand the formula for the sum of n positive integers from here. I understand how the binomial part works.
For the expansion of $(k-1)^2$:
$(k−1)^2 = k^2 − 2k + 1$
Rearrange the terms as below:
$k^2−(k−1)^2=2k−1$
Now sum both sides:
$$\sum_{k=1}^n (k^2−(k−1)^2) = 2\sum_{k=1}^nk-\sum_{k=1}^n1$$
Problem and my question starts from here, Can anyone please add in simple words with simple equation/calculation how below results come:
It equals $n^2$.The right side equals $2S_n$$ - n$, which gives $2S_n$$-n$=$n^2$,so $S_n$ = $\frac{n(n+1)}2$.