[nooonuii]

$\dfrac{1}{2[\sqrt{1}]+1}+\dfrac{1}{2[\sqrt{2}]+1}+\dfrac{1}{2[\sqrt{3}]+1}+\cdots+\dfrac{1}{2[\sqrt{100}]+1}=\dfrac{2^2-1^2}{2(1)+1}+\dfrac{3^2-2^2}{2(2)+1}+\dfrac{4^2-3^2}{2(3)+1}+\cdots+\dfrac{10^2-9^2}{2(9)+1}+\dfrac{1}{2(10)+1}$