Question

Calculati: 1^2+5^2+9^2+13^2+17^2+...+201^2;
^2=la puterea a doua

Answer 1

1² =(1+0·2²)² =1
5² = (1+1·2²)²= 1 +2·1·2² +(1·2²)²
9² =(1+2·2²)² = 1+ 2·2·2² +(2·2²)²
13 = (1+3·2²)² = 1+2·3·2² +(3·2²)²
17 = (1+4·2²)² = 1+2·4·2² +(4·2²)²
......................................................
201² = (1+50·2²)² = 1 +2·50·2² +(50·2²)²
S = 1×51 +2·2²(1+2+3+......+50) + (2²)² ·(1² + 2² +3² +.............+ 50² ) =
= 51+ 8· 50·51/2 + 16·50·51·101/6 =51·201 +400·17·101 = ·17(603+40400) =
S = 17·41003 = 697051
1²+2² +3² +.........+n² = n(n+1)(2n+1)/6