Question

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

• A. $\frac{56}{111}$
• B. $\frac{58}{111}$
• C. $\frac{55}{111}$
• D. $\frac{59}{111}$

Answer 1

Answer: (C)

$\frac{55}{111}$

Answer 2

Tr​=2(r4+r2+1)(r2+r+1)−(r2−r+1)​
⇒Tr​=21​[r2−r+11​−r2+r+11​]
T1​=21​[11​−31​]
T2​=21​[31​−71​]
T3​=21​[71​−131​]
⋮
T10​=21​[911​−1111​]
⇒r=1∑10​Tr​=21​[1−1111​]=11155​