Question

The sum of the series ${\frac{1}{1\times 2}}^{25}{C}_0+{\frac{1}{2\times 3}}^{23}{C}_1+{\frac{1}{3\times 4}}^{25}{C}_2+......+{\frac{1}{26\times 27}}^{25}{C}_{25}$

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

Answer 1

The correct option is B $\frac{2^{27}-28}{26\times 27}$

$(1+x{)}^{25}={}^{25}{C}_0+{}^{25}{C}_{1}x+{}^{25}{C}_2{x}^2+....+{}^{25}{C}_{25}{x}^{25}-------\left(1\right)$

Intergrating eq (1) w.r.t $x$ with limit $0$ to $x$

${\int }_0^x(1+x{)}^{25}={\int }_0^x({}^{25}{C}_0+{}^{25}{C}_{1}x+{}^{25}{C}_2{x}^2+....+{}^{25}{C}_{25}{x}^{25})$

$\Rightarrow \frac{(1+x{)}^{26}-1}{26}={}^{25}{C}_{0}x+\frac{{}^{25}{C}_1{x}^2}{2}+\frac{{}^{25}{C}_2{x}^3}{3}+....+\frac{{}^{25}{C}_{25}{x}^{26}}{26}----\left(2\right)$

Intergrating eq (2) w.r.t $x$ with limit $0$ to $1$ , we get

${[\frac{{}^{25}{C}_0{x}^2}{2}+\frac{{}^{25}{C}_1{x}^3}{2\times 3}+\frac{{}^{25}{C}_2{x}^4}{3\times 4}+....+\frac{{}^{25}{C}_{25}{x}^{27}}{26\times 27}]}_0^1={[\frac{(1+x{)}^{27}}{26\times 27}-\frac{x}{26}]}_0^1$

$\frac{{}^{25}{C}_0}{1\times 2}+\frac{{}^{25}{C}_1}{2\times 3}+\frac{{}^{25}{C}_2}{3\times 4}+....+\frac{{}^{25}{C}_{25}}{26\times 27}=\frac{(2{)}^{27}-1}{26\times 27}-\frac{1}{26}$

$\frac{{}^{25}{C}_0}{1\times 2}+\frac{{}^{25}{C}_1}{2\times 3}+\frac{{}^{25}{C}_2}{3\times 4}+....+\frac{{}^{25}{C}_{25}}{26\times 27}=\frac{2^{27}-28}{26\times 27}$