Question

Tính:

\(\left(\frac{1}{3^2}-1\right).\left(\frac{1}{4^2}-1\right).\left(\frac{1}{5^2}-1\right)...\left(\frac{1}{50^2}-1\right)\)

Answer 1

\(\left(\frac{1}{3^2}-1\right).\left(\frac{1}{4^2}-1\right).\left(\frac{1}{5^2}-1\right)...\left(\frac{1}{50^2}-1\right)\)

\(=\frac{-8}{3^2}.\frac{-15}{4^2}.\frac{-24}{25}...\frac{-2499}{50^2}\)

\(=\frac{8}{3^2}.\frac{15}{4^2}.\frac{24}{5^2}...\frac{2499}{50^2}\) (vì có 48 thừa số âm nên kết quả là dương)

\(=\frac{2.4}{3.3}.\frac{3.5}{4.4}.\frac{4.6}{5.5}...\frac{49.51}{50.50}\)

\(=\frac{2.3.4...49}{3.4.5...50}.\frac{4.5.6...51}{3.4.5...50}\)

\(=\frac{2}{50}.\frac{51}{3}\)

\(=\frac{1}{25}.17=\frac{17}{25}\)