Sequences & Series
Infinite AGP — Finding k
nta_pyq_2023_apr
Grade 11
Question:
For $k\in\mathbb{N}$, if $1+\dfrac{4}{k}+\dfrac{8}{k^2}+\dfrac{13}{k^3}+\dfrac{19}{k^4}+\cdots=10$, then $k$ is equal to
Step-by-Step Solution
Key Concept: Let $S=\sum a_r/k^{r-1}$. Differences of numerators: $4,4,5,6,7,\ldots$ — secondary differences constant at 1 after 2nd. Use $S-S/k$ repeatedly.
$k=2$.
Correct Answer: 2