Question

The sum of the series \[ \frac{1}{1 - 3 \cdot 1^2 + 1^4} + \frac{2}{1 - 3 \cdot 2^2 + 2^4} + \frac{3}{1 - 3 \cdot 3^2 + 3^4} + \ldots \text{ up to 10 terms} \] is

• A. \( \frac{45}{109} \)
• B. \( -\frac{45}{109} \)
• C. \( \frac{55}{109} \)
• D. \( -\frac{55}{109} \)

Answer 1

The Correct Option is D

General term of the sequence,

$T_r = \frac{r}{1 - 3r^2 + r^4}$

$T_r = \frac{r}{r^4 - 3r^2 + 1 - r^2}$

$T_r = \frac{r}{(r^2 - 1)^2 - r^2}$

$T_r = \frac{r}{(r^2 - r - 1)(r^2 + r - 1)}$

$T_r = \frac{1}{2} \left[ \frac{1}{(r^2 - r - 1)} - \frac{1}{(r^2 + r - 1)} \right]$

Sum of 10 terms,

$\sum_{r=1}^{10} T_r = \frac{1}{2} \left[ \frac{1}{-1} - \frac{1}{109} \right] = \frac{55}{109}$