Sequence and Series
Telescoping Series
JEE Main 2022
Grade 11
Question:
If $\sum_{k=1}^{10} \frac{k}{k^4 + k^2 + 1} = \frac{m}{n}$ where $\gcd(m, n) = 1$, find $m + n$.
Step-by-Step Solution
Key Concept: Factor: $k^4 + k^2 + 1 = (k^2 + k + 1)(k^2 - k + 1)$. Use partial fractions to write each term as $\frac{1}{2} \left[\frac{1}{k^2 - k + 1} - \frac{1}{k^2 + k + 1}\right]$ and telescope.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: 166