Question

1/2+1/6+1/12+1/20+...........+1/x(x+1)= 200/201

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Answer 1

\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+....+\frac{1}{x\left(x+1\right)}=\frac{200}{201}\)

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{x\left(x+1\right)}=\frac{200}{201}\)

\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{200}{201}\)

\(1-\frac{1}{x+1}=\frac{200}{201}\)

=> \(\frac{1}{x+1}=1-\frac{200}{201}=\frac{1}{201}\)

=> x + 1 = 201

=> x = 201 - 1

=> x = 200