Sequences & Series
Telescoping Sum — p+q
nta_pyq_2026_jan
Grade 11
Question:
If $\displaystyle\sum_{r=1}^{25}\left(\dfrac{r}{r^4+r^2+1}\right)=\dfrac{p}{q}$, where $p$ and $q$ are positive integers such that $\gcd(p,q)=1$, then $p+q$ is equal to _____.
Step-by-Step Solution
Key Concept: Partial fractions: $\tfrac{r}{r^4+r^2+1}=\tfrac{r}{(r^2+r+1)(r^2-r+1)}=\tfrac{1}{2}\left(\tfrac{1}{r^2-r+1}-\tfrac{1}{r^2+r+1}\right)$. Telescoping.
$\tfrac{p}{q}=\tfrac{325}{651}$. $p+q=976$.
Correct Answer: 976