The value of 12 – 22 + 32 – 42 + ..... + 112 is equal to(a) 55 (b

1.	The value of 12 – 22 + 32 – 42 + ..... + 112 is equal to(a) 55 (b) 66 (c) 77 (d) 88
Answer» (b) 66. Let S = 1² – 2² + 3² – 4² + ..... + 11² = (1² + 2² + 3² + 4² + ..... + 11²) – 2(2² + 4² + 6² + 8² + 11²) = (1² + 2² + 3² + 4² + ..... + 11²) – 2³(1² + 2² + 3² + 4² + 5²) = \(\frac{11\times(11+1)\times(2\times11\times1)}{6}\) - \(\frac{8\times5\times(5+1)\times(2\times5+1)}{6}\) = \(\frac{11\times12\times23}{6}\) - \(\frac{8\times5\times6\times11}{6}\) = 22 x 23 - 40 x 11 = 506 – 440 = 66.

The value of 12 – 22 + 32 – 42 + ..... + 112 is equal to(a) 55 (b) 66 (c) 77 (d) 88

Answer»

(b) 66.

Let S = 1² – 2² + 3² – 4² + ..... + 11²

= (1² + 2² + 3² + 4² + ..... + 11²) – 2(2² + 4² + 6² + 8² + 11²)

= (1² + 2² + 3² + 4² + ..... + 11²) – 2³(1² + 2² + 3² + 4² + 5²)

= \(\frac{11\times(11+1)\times(2\times11\times1)}{6}\) - \(\frac{8\times5\times(5+1)\times(2\times5+1)}{6}\)

= \(\frac{11\times12\times23}{6}\) - \(\frac{8\times5\times6\times11}{6}\) = 22 x 23 - 40 x 11

= 506 – 440 = 66.