Question

Cu cat este egal 1*3+2*4+3*5...+n(n+2) ?

Answer 1

Suma data este:

$S=1\cdot(1+2)+2\cdot(2+2)+3\cdot(3+2)+...+n(n+2)=$

$1^2+1\cdot2+2^2+2\cdot2+3^2+3\cdot2+...+n^2+n\cdot2=$

$1^2+2^2+3^2+...+n^2+2(1+2+3+...+n)=$

$\dfrac{n(n+1)(2n+1)}{6}+2\cdot\dfrac{n(n+1)}{2}=\dfrac{n(n+1)(2n+7)}{6}$