Question

The sum of the series ${\frac{1}{1\times 2}}^{{}^{25}}{C}_0+{\frac{1}{2\times 3}}^{{}^{25}}{C}_1+{\frac{1}{3\times 4}}^{{}^{25}}{C}_2+.........+{\frac{1}{26\times 27}}^{{}^{25}}{C}_{25}$

• A. $\frac{2^{27}-1}{26\times 27}$
• B. $\frac{2^{27}-28}{26\times 27}$
• C. $\frac{1}{2}\left(\frac{2^{26}+1}{2627}\right)$
• D. $\frac{2^{26}-1}{52}$

Answer 1

Answer: (B)

Given series is,
$\frac{1}{1\times 2}{}^{25}{C}_0+\frac{1}{2\times 3}{}^{25}{C}_1+\frac{1}{3\times 4}{}^{25}{C}_2$
$+\dots +\frac{1}{26\times 27}{}^{25}{C}_{25}$
$\because {\int }_0^x(1+x{)}^{25}dx={\int }_0^x[{}^{25}{C}_0+{}^{25}{C}_{1}x$
$+{}^{25}{C}_2{x}^2+\dots +{}^{25}{C}_{25}{x}^{25}]dx$
On integrating w.r.t. $x$, taking limits 0 to $x$, we get
${\left[\frac{(1+x{)}^{26}}{26}\right]}_0^x$
$={\left[{}^{25}{C}_{0}x+{}^{25}{C}_1\cdot \frac{x^2}{2}+{}^{25}{C}_2\frac{x^3}{3}+\dots +{}^{25}{C}_{25}\cdot \frac{x^{26}}{26}\right]}_0^x$
$\Rightarrow \left\{\frac{1}{26}(1+x{)}^{26}-\frac{1}{26}\right\}$
$={}^{25}{C}_{0}x+{}^{25}{C}_1\cdot \frac{x^2}{2}+\dots +{}^{25}{C}_{25}\cdot \frac{x^{26}}{26}$
Again, integrating w.r.t. $x$, taking limits 0 to 1 , we get
$\frac{1}{26}{\int }_0^1[1+x{)}^{26}-1]dx$
$={\int }_0^1\left[{}^{25}{C}_{0}x+{}^{25}{C}_1\cdot \frac{x^2}{2}+\dots +{}^{25}{C}_{25}\frac{x^{26}}{26}\right]dx$
$\Rightarrow \frac{1}{26}{\left[\frac{(1+x{)}^{27}}{27}-x\right]}_0^1$
$={\left[{}^{25}{C}_0\frac{x^2}{2}+{}^{25}{C}_1\cdot \frac{x^3}{2\times 3}+\dots +{}^{25}{C}_{25}\frac{x^{27}}{26\times 27}\right]}_0^1$
$\Rightarrow \frac{1}{26}\left\{\frac{2^{27}}{27}-1-\frac{1}{27}\right\}=\frac{1}{2}{}^{25}{C}_0+\frac{1}{2\times 3}\cdot {}^{25}{C}_1$
$+\dots +\frac{1}{26\times 27}\cdot {}^{25}{C}_{25}$
$\therefore \frac{1}{1\times 2}\cdot {}^{25}{C}_0+\frac{1}{2\times 3}{}^{25}{C}_1+\frac{1}{3\times 4}{}^{25}{C}_2$
$+\dots +\frac{1}{26\times 27}\cdot {}^{25}{C}_{25}=\frac{2^{27}-28}{26\times 27}$