Question

ne ajudem com urgencia ate 9 :20 por favor utiize o principio de indução
n
Σ =1 i²       n(n+1)(2n+1)
i                  __________
                            6

Answer 1

$\displaystyle S_n=\sum\limits_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6}\hspace{3mm}\cdots\cdots \textcircled{1}\\ \\ \\ S_{n+1}=\sum\limits_{i=1}^{n+1}i^2\\ \\ S_{n+1}=\sum\limits_{i=1}^{n}i^2+(n+1)^2\\ \\ \text{Por }\textcircled{1}:\\ S_{n+1}=\frac{n(n+1)(2n+1)}{6}+(n+1)^2\\ \\ S_{n+1}=(n+1)\left[\frac{n(2n+1)}{6}+(n+1)\right]$

$\displaystyle S_{n+1}=(n+1)\left[\frac{2n^2+7n+6}{6}\right]\\ \\ S_{n+1}=(n+1)\left[\frac{(n+2)(2n+3)}{6}\right]\\ \\ \\ \boxed{S_{n+1}=\frac{(n+1)\left[(n+1)+1\right]\left[2(n+1)+1\right]}{6}}$