Sequences & Series
Sum Formula — α+β from Given Conditions
nta_pyq_2026_jan
Grade None
Question:
Let $\displaystyle\sum_{k=1}^n a_k=\alpha n^2+\beta n$. If $a_{10}=59$ and $a_6=7a_1$, then $\alpha+\beta$ is equal to
Step-by-Step Solution
Key Concept: $a_n=\alpha(2n-1)+\beta$ for $n\geq2$ (and $a_1=\alpha+\beta$). $a_6=11\alpha+\beta$, $a_{10}=19\alpha+\beta=59$. $a_6=7a_1\Rightarrow11\alpha+\beta=7(\alpha+\beta)\Rightarrow4\alpha=6\beta$.
$\alpha=3$, $\beta=2$. $\alpha+\beta=5$.
Correct Answer: 4