Sequences & Series
Sum Involving nth Term of Sequence — Prime Factorization
nta_pyq_2023_apr
Grade 11
Question:
Let $\langle a_n\rangle$ with $\sum_{k=1}^n a_k=\frac{n^2+3n}{(n+1)(n+2)}$. If $28\sum_{k=1}^{10}\frac{1}{a_k}=p_1p_2\cdots p_m$, then $m$ is equal to
Step-by-Step Solution
Key Concept: $a_n=S_n-S_{n-1}=\frac{4}{n(n+1)(n+2)}$. $\frac{1}{a_n}=\frac{n(n+1)(n+2)}{4}$. Use telescoping: $\sum_{k=1}^{10}\frac{1}{a_k}=\frac{1}{16}\cdot10\cdot11\cdot12\cdot13$.
$m=6$.
Correct Answer: 3