Question

What is the sum of the series 1² – 2² + 3² – 4² + ...... + 99² – 100².

Answer 1

1² – 2² + 3² – 4² + 5² – 6² + ...... + 99² – 100² 

= (1 – 2) (1 + 2) + (3 – 4) (3 + 4) + (5 – 6) (5 + 6) + ...... + (99 – 100) (99 + 100) 

= – (1 + 2) – (3 + 4) – (5 + 6) ..... – (99 + 100) 

= – (1 + 2 + 3 + 4 + 5 + 6 + ..... + 99 + 100) 

= – \((\frac{100}{2} (1+100))\) (∵ S_n = \(\frac{n}{2}\) (first term + last term)) 

= – 50 × 101 = – 5050.