TH1: n lẻ
=>n+1 chẵn
=>\(\left(-1\right)^{n+1}\) chẵn
=>n=2k+1
\(S=1^2-2^2+3^2-4^2+\cdots+\left(-1\right)^{n+1}\cdot n^2\)
\(=1^2-2^2+3^2-4^2+\cdots+\left(2k+1\right)^2\)
\(=1^2-2^2+3^2-4^2+\cdots+\left(2k-1\right)^2-\left(2k\right)^2+\left(2k+1\right)^2\)
\(=\left(1-2\right)\left(1+2\right)+\left(3+4\right)\left(3-4\right)+\cdots+\left(2k-1-2k\right)\left(2k-1+2k\right)+\left(2k+1\right)^2\)
\(=-\left(1+2+3+\cdots+2k-1+2k\right)+\left(2k+1\right)^2\)
\(=-\frac{2k\left(2k+1\right)}{2}+\left(2k+1\right)^2=-k\left(2k+1\right)+\left(2k+1\right)^2\)
=(2k+1)(-k+2k+1)
=(2k+1)(k+1)
\(=2k^2+3k+1\)
TH2: n chẵn
=>n+1 lẻ
=>n=2k
\(S=1^2-2^2+3^2-4^2+\cdots+\left(-1\right)^{n+1}\cdot n^2\)
\(=1^2-2^2+3^2-4^2+\cdots-\left(2k\right)^2\)
\(=\left(1-2\right)\left(1+2\right)+\left(3-4\right)\left(3+4\right)+\cdots+\left(2k-1-2k\right)\left(2k-1+2k\right)\)
=-(1+2+...+2k-1+2k)
\(=\frac{-2k\left(2k+1\right)}{2}=-k\left(2k+1\right)\)
