2702.11 – Add'Em Up

Evaluate the sum:

12−22+32−42+52−62+⋯+1992.1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + \cdots + 199^2.

Solution

Pair 'em up:

12+(32−22)+(52−42)+(72−62)+…+(1992−1982)=1+5+9+13+⋯+397.1^2 + (3^2- 2^2) + (5^2- 4^2) + (7^2 - 6^2) + \ldots + (199^2 - 198^2) = 1 + 5 + 9 + 13 + \cdots + 397.

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

(number of terms) ⋅ (average of the first and last terms)=992(1+397)=19701. \text{(number of terms) } \cdot \text{ (average of the first and last terms)} = \frac{99}{2} (1 + 397) = 19701.