Sequences And Series Question 18
Question 18 - 01 February - Shift 1
The sum to 10 terms of the series
$\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots$ is :-
(1) $\frac{59}{111}$
(2) $\frac{55}{111}$
(3) $\frac{56}{111}$
(4) $\frac{58}{111}$
Show Answer
Answer: (2)
Solution:
Formula: General term ( $n^{\text {th }}$ term) of an A.G.P.
$ \begin{aligned} & T_r=\frac{(r^{2}+r+1)-(r^{2}-r+1)}{2(r^{4}+r^{2}+1)} \\ & \Rightarrow T_r=\frac{1}{2}[\frac{1}{r^{2}-r+1}-\frac{1}{r^{2}+r+1}] \\ & T_1=\frac{1}{2}[\frac{1}{1}-\frac{1}{3}] \\ & T_2=\frac{1}{2}[\frac{1}{3}-\frac{1}{7}] \\ & T_3=\frac{1}{2}[\frac{1}{7}-\frac{1}{13}] \\ & \text{ : } \\ & T _{10}=\frac{1}{2}[\frac{1}{91}-\frac{1}{111}] \\ & \Rightarrow \sum _{r=1}^{10} T_r=\frac{1}{2}[1-\frac{1}{111}]=\frac{55}{111} \end{aligned} $
