\(100^2-99^2+98^2-97^2+......+4^2-3^2+2^2-1^2\)
\(=\left(100^2-99^2\right)+\left(98^2-97^2\right)+......+\left(4^2-3^2\right)+\left(2^2-1^2\right)\)
\(=\left(100-99\right)\left(100+99\right)+\left(98-97\right)\left(98+97\right)+......+\left(4-3\right)\left(4+3\right)+\left(2-1\right)\left(2+1\right)\)
\(=1.\left(100+99\right)+1.\left(98+97\right)+......+1.\left(4+3\right)+1.\left(2+1\right)\)
\(=199+195+......+7+3\)
\(=\dfrac{\left(199+3\right)\left[\left(199-3\right):4+1\right]}{2}\)
\(=\dfrac{202.\left(196:4+1\right)}{2}\)
\(=\dfrac{202.\left(49+1\right)}{2}\)
\(=\dfrac{202.50}{2}\)
\(=\dfrac{10100}{2}\)
\(=5050\)
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