Evaluate 6²-7²+8²-9² + ... + 998²-999²

-499,485.

Step-by-step explanation:

We can transform this to an arithmetic series by working it out in pairs:

6^2 - 7^2 = (6-7) (6+7) = - 13

8^2 - 9^2 = (8-9) * 8+9) = - 17

10^2 - 11^2 = - 1 * 21 = - 21 and so on

The common difference is - 4.

The number of terms in this series is (998 - 6) / 2 + 1

= 992/2 + 1 = 497.

Sum of n terms of an A. S:

= n/2 [2a1 + (n - 1) d

Here a1 = - 13, n = 497, d = - 4:

Sum = (497/2) [-26 - 4 (497-1) ]

= 497/2 * - 2010

= - 499,485.