Question

The sum of the series $\frac{1}{1\times2} ^{^{25}}C_{0} + \frac{1}{2\times3}^{^{25}}C_{1} + \frac{1}{3\times4} ^{^{25}}C_{2} + ......... + \frac{1}{26\times27} ^{^{25}}C_{25}$

• A. $\frac{2^{27}-1}{26\times27}$
• B. $\frac{2^{27}-28}{26\times27}$
• C. $\frac{1}{2}\left(\frac{2^{26}+1}{26\,\,27}\right)$
• D. $\frac{2^{26}-1}{52}$

Answer 1

Answer: (B)

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