Question
The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If r 1 is the common ratio, then what is the number of possible real values of r?
NDA
Mathematics
Sequences and Series
2025
NDA 2025 (Phase 1)
NDA Mathematics Question (2025) — Solution
Options
- A. One
- B. Two
- C. Three
- D. More than three
Answer
B. Two
Step-by-step solution
Let the first term of the GP be a and the common ratio be r. The sum of the first n terms is given by S_n = a(r^n - 1) r - 1 . According to the given condition, S_8 = 5S_4. a(r^8 - 1) r - 1 = 5 a(r^4 - 1) r - 1 For a non-zero first term a and given r 1: r^8 - 1 = 5(r^4 - 1) (r^4 - 1)(r^4 + 1) = 5(r^4 - 1) Assuming r^4 - 1 0 to avoid the trivial zero-sum case: r^4 + 1 = 5 r^4 = 4 Since r is a real number, r^2 must be positive. r^2 = 2 r = 2 Therefore, there are two possible real values of r.
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