Question

Chứng minh:

S\(_1\)=\(\dfrac{1}{1.2}\)+\(\dfrac{1}{2.3}\)+\(\dfrac{1}{3.4}\)+...+\(\dfrac{1}{99.100}\)<1

Answer 1

S₁=\(\dfrac{1}{1-2}+\dfrac{1}{2-3}+\dfrac{1}{3-4}+...+\dfrac{1}{99-100}\)

S₁=1-\(\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-...-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{100}\)

S₁=\(\dfrac{99}{100}\)<1(đpcm)

Answer 2

S₁=\(\dfrac{1}{1-2}+\dfrac{1}{2-3}+\dfrac{1}{3-4}+...+\dfrac{1}{99-100}\)

S₁=1-\(\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-...-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{100}\)

S₁=\(\dfrac{99}{100}\)<1(đpcm)