Sequences & Series
AP — Sum of Squares Relation
nta_pyq_2024_apr
Grade None
Question:
Let $a_1,a_2,a_3,\ldots$ be in an arithmetic progression of positive terms. Let $A_k=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2k-1}^2-a_{2k}^2$. If $A_3=-153$, $A_5=-435$ and $a_1^2+a_2^2+a_3^2=66$, then $a_{17}-A_7$ is equal to
Step-by-Step Solution
Key Concept: $A_k=\sum_{j=1}^{k}(a_{2j-1}^2-a_{2j}^2)=\sum_{j=1}^{k}(a_{2j-1}-a_{2j})(a_{2j-1}+a_{2j})$. With common difference $d$: each pair gives $(-d)(a_{2j-1}+a_{2j})$.
$a_{17}-A_7=910$.
Correct Answer: 910