2702.11 – Add'Em Up

Evaluate the sum:

12−22+32−42+52−62+⋯+19921^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + \cdots + 199^2

Solution

Pair 'em up:

  12+(32−22)+(52−42)+⋯+(1992−1982)=1+5+9+13+⋯+397\begin{align*}&\quad \, \, 1^2 + (3^2- 2^2) + (5^2- 4^2) + \cdots + (199^2 - 198^2) \\&= 1 + 5 + 9 + 13 + \cdots + 397\end{align*}

This is an arithmetic progression with 1st term 1, difference 4, and last term 397. The sum of these progressions is

  (# of terms)×(average of the first and last terms)=992(1+397)=19701\begin{align*}&\quad \, \, \text{(\# of terms)} \\&\times \text{(average of the first and last terms)} \\\hline \\[-1em]&= \dfrac{99}{2} (1 + 397) \\[1em]&= 19701\end{align*}