Question

1/1×2+1/2×3+1/3×4••••+1/99×100

Answer 1

\(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+...+\dfrac{1}{99\times100}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\)

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

\(=1-\dfrac{1}{100}-\dfrac{99}{100}\)