Sequences & Series
Sum of series with denominator involving squares
nta_pyq_2023_jan
Grade 11
Question:
The sum to 10 terms of the series $\dfrac{1}{1 + 1^2 + 1^4} + \dfrac{2}{1 + 2^2 + 2^4} + \dfrac{3}{1 + 3^2 + 3^4} + \ldots$ is:
\dfrac{59}{111}
\dfrac{55}{111}
\dfrac{56}{111}
\dfrac{58}{111}
Step-by-Step Solution
Key Concept: The general term is $\frac{n}{1+n^2+n^4} = \frac{n}{(n^2+n+1)(n^2-n+1)}$. Use partial fractions: $= \frac{1}{2}\left(\frac{1}{n^2-n+1} - \frac{1}{n^2+n+1}\right)$ (telescoping).
$S_{10} = \frac{1}{2}\left(\frac{1}{1} - \frac{1}{111}\right) = \frac{55}{111}$.
Correct Answer: 2