Sequences And Series Question 16
Question 16 - 2024 (31 Jan Shift 1)
The sum of the series $\frac{1}{1-3 \cdot 1^{2}+1^{4}}+\frac{2}{1-3 \cdot 2^{2}+2^{4}}+\frac{3}{1-3 \cdot 3^{2}+3^{4}}+\ldots$ up to 10 terms is
(1) $\frac{45}{109}$
(2) $-\frac{45}{109}$
(3) $\frac{55}{109}$
(4) $-\frac{55}{109}$
Show Answer
Answer (4)
Solution
General term of the sequence,
$$ \begin{aligned} & T _r=\frac{r}{1-3 r^{2}+r^{4}} \\ & T _r=\frac{r}{r^{4}-2 r^{2}+1-r^{2}} \\ & T _r=\frac{r}{\left(r^{2}-1\right)^{2}-r^{2}} \\ & T _r=\frac{r}{\left(r^{2}-r-1\right)\left(r^{2}+r-1\right)} \\ & T _r=\frac{\frac{1}{2}\left[\left(r^{2}+r-1\right)-\left(r^{2}-r-1\right)\right]}{\left(r^{2}-r-1\right)\left(r^{2}+r-1\right)} \\ & =\frac{1}{2}\left[\frac{1}{r^{2}-r-1}-\frac{1}{r^{2}+r-1}\right] \end{aligned} $$
Sum of 10 terms,
$\sum _{r=1}^{10} T _r=\frac{1}{2}\left[\frac{1}{-1}-\frac{1}{109}\right]=\frac{-55}{109}$
