Sequences & Series
GP — Common Ratio from Sum Condition
nta_pyq_2024_jan
Grade 11
Question:
If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to
Step-by-Step Solution
Key Concept: Sum of all 64 terms $=S_{64}=a\frac{1-r^{64}}{1-r}$. Odd-numbered terms (1st, 3rd, ..., 63rd) form a GP with first term $a$, ratio $r^2$, 32 terms: sum $=a\frac{1-r^{64}}{1-r^2}$. Condition: $S_{64}=7\times\text{(sum of odd terms)}$.
$1+r=7\Rightarrow r=6$.
Correct Answer: 4