Question

tinh D=(1/2²-1)(1/3²-1)(1/4²-1)...(1/100²-1)

dung minh se cho 4 tick nha nhanh len day

Answer 1

\(D=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{100^2}-1\right)\)

\(=-\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)....\left(1-\frac{1}{100^2}\right)\)

\(=-\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}.\frac{4^2-1}{4^2}.....\frac{100^2-1}{100^2}\)

\(=-\left(\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}....\frac{99.101}{100^2}\right)\)

\(=-\left(\frac{1.2.3...99}{2.3.4...100}.\frac{3.4.5...101}{2.3.4....100}\right)\)

\(=-\left(\frac{1}{100}.\frac{101}{2}\right)\)

\(=-\frac{101}{200}\)

Answer 2

\(D=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\cdot\cdot\cdot\left(\frac{1}{100^2}-1\right)\)(có 50 số hạng)

\(\Rightarrow D=\left(\frac{2^2-1}{2^2}\right)\left(\frac{3^2-1}{3^2}\right)\cdot\cdot\cdot\left(\frac{100^2-1}{100^2}\right)\)

\(\Rightarrow D=\frac{1\cdot3}{2^2}\cdot\frac{2\cdot4}{3^2}\cdot\cdot\cdot\frac{99\cdot101}{100^2}\)

\(\Rightarrow D=\frac{101}{2\cdot100}=\frac{101}{200}\)