Question

11²+12²+13²+14²+15²+16²+17²+18²+19²+20²=​

Answer 1

11² + 12² + 13² + ... + 20² = 2485

we have to evaluate,

11² + 12² + 13² + ..... + 20²

it can be written as,

1² + 2² + 3² + ... + 20² - (1² + 2² + 3² + .... + 10²)

we know, 1² + 2² + 3² + 4² + .... + n² = n(n + 1)(2n + 1)/6

so, 1² + 2² + 3² + .... + 20² = 20(20 + 1)(2 × 20 + 1)/6

= 20 × 21 × 41/6 = 70 × 41 = 2870

similarly, 1² + 2² + 3² + .... + 10² = 10 (10 + 1)(2 × 10 + 1)/6 = 10 × 11 × 21/6 = 35 × 11 = 385

then, 1² + 2² + 3² + ... + 20² - (1² + 2² + 3² + .... + 10²) = 2870 - 385 = 2485

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