Question

Find the value of
$(1-\frac{1}{2^2})(1-\frac{1}{3^3})(1-\frac{1}{4^2})(1-\frac{1}{5^2})........(1-\frac{1}{9^2})(1-\frac{1}{{10}^2})$

• A. $\frac{5}{12}$
• B. $\frac{1}{2}$
• C. $\frac{11}{20}$
• D. $\frac{7}{10}$

Answer 1

Answer: (C)

$\frac{11}{20}$

Given expression can be expanded as $(1-\frac{1}{4})(1-\frac{1}{9})(1-\frac{1}{16})(1-\frac{1}{25})$............$(1-\frac{1}{81})(1-\frac{1}{100})$
$=\frac{3^1}{4_1}\times \frac{8^{2^1}}{9_{8_1}}\times \frac{{15}^{5^1}}{{25}_{5_1}}\times \frac{{24}^{3^1}}{{36}_{{12}_1}}\times \frac{{48}^{4^1}}{{49}_{7_1}}$
$=\frac{{63}^{7^1}}{{64}_{{16}^1}}\times \frac{{80}^{5^1}}{{81}_{9_1}}\times \frac{{99}^{11}}{{100}_{20}}=\frac{11}{20}$