Consider the following expression :

1n+1+2n+1+3n+1+...+nn+1

=n(n+1)2n+1=n2

So we have :

12+(13+23)+(14+24+34)+(15+25+35+45)+...+(1100+2100+3100+...+99100)

=12+22+32+42+...+992

=99.10022=99.1004=99.25=2475

Consider the following expression :

\(\dfrac{1}{n+1}+\dfrac{2}{n+1}+\dfrac{3}{n+1}+...+\dfrac{n}{n+1}\)

\(=\dfrac{\dfrac{n\left(n+1\right)}{2}}{n+1}=\dfrac{n}{2}\)

So we have :

\(\dfrac{1}{2}+\left(\dfrac{1}{3}+\dfrac{2}{3}\right)+\left(\dfrac{1}{4}+\dfrac{2}{4}+\dfrac{3}{4}\right)+\left(\dfrac{1}{5}+\dfrac{2}{5}+\dfrac{3}{5}+\dfrac{4}{5}\right)+...+\left(\dfrac{1}{100}+\dfrac{2}{100}+\dfrac{3}{100}+...+\dfrac{99}{100}\right)\)

\(=\dfrac{1}{2}+\dfrac{2}{2}+\dfrac{3}{2}+\dfrac{4}{2}+...+\dfrac{99}{2}\)

\(=\dfrac{\dfrac{99.100}{2}}{2}=\dfrac{99.100}{4}=99.25=2475\)