Question

Find the sum of the series:$\frac{3}{1.2}\left(\frac{1}{2}\right)+\frac{4}{2.3}{\left(\frac{1}{2}\right)}^2+\frac{5}{3.4}{\left(\frac{1}{2}\right)}^3+........$ to n terms

• A. $2-\frac{1}{(n+1)2^n}$
• B. $1-\frac{1}{(n+1)2^n}$
• C. $\frac{1}{(n+1)2^n}$
• D. $n-\frac{1}{(n+1)2^n}$
  

Answer 1

The correct option is B $1-\frac{1}{(n+1)2^n}$

$\Rightarrow \frac{3}{1.2}\left(\frac{1}{2}\right)+\frac{4}{2.3}{\left(\frac{1}{2}\right)}^2+\frac{5}{3.4}{\left(\frac{1}{2}\right)}^3+..........+n$ terms

$\Rightarrow t_n=\frac{n+2}{n(n+1)}.{\left(\frac{1}{2}\right)}^m$

$=\frac{2n+2-n}{n(n+1)}.{\left(\frac{1}{2}\right)}^m=\frac{2(n-1)-n}{n(n+1)}{\left(\frac{1}{2}\right)}^n$

$=\frac{1}{n}{\left(\frac{1}{2}\right)}^{n-1}-\frac{1}{n+1}{\left(\frac{1}{2}\right)}^n$

$S_n=\sum _{n=1}^n{t}_n$

$=\{\frac{1}{1}{\left(\frac{1}{2}\right)}^0-\frac{1}{2}{\left(\frac{1}{2}\right)}^1\}+\{\frac{1}{1}{\left(\frac{1}{2}\right)}^1-\frac{1}{2}{\left(\frac{1}{2}\right)}^2\}+............+\{\frac{1}{n}{\left(\frac{1}{2}\right)}^{n-1}-\frac{1}{n+1}{\left(\frac{1}{2}\right)}^n\}$

$=1-\frac{1}{(n+1)}{\left(\frac{1}{2}\right)}^n$

$=1-\frac{1}{(n+1)2^n}$ Ans