Question

S= 100 ^ 2 - 99 ^ 2 + 98 ^ 2 - 97 ^ 2 +…+2^ 2 -1^ 2

Answer 1

\(S=100^2-99^2+98^2-97^2+...+2^2-1^2\)

\(S=\left(100^2-99^2\right)+\left(98^2-97^2\right)+...+\left(2^2-1^2\right)\)

\(S=\left(100-99\right)\left(99+100\right)+\left(98-97\right)\left(98+97\right)+...+\left(2-1\right)\left(2+1\right)\)

\(S=1\cdot199+1\cdot195+1\cdot193+...+1\cdot3\)

\(S=199+195+193+...+3\)

\(S=\left(3+199\right)\cdot\left[\left(199-3\right):2+1\right]:2\)

\(S=202\cdot99:2\)

\(S=101\cdot99\)

\(S=9999\)