Question

For a geometric progression, S₄ = 10 S₂. Find the common ratio.

Answer 1

S₄ = 10 S₂

S_n = \(\frac{a[r^n−1]}{r−1]}\)

S₄ = \(\frac{a[r^4−1]}{r−1]}\) and S₂ = \(\frac{a[r^2−1]}{r−1]}\)

Now, S₄ = 10 S₂

\(\frac{a[r^4−1]}{r−1]}\) = 10 × \(\frac{a[r^2−1]}{r−1]}\)

∴ r⁴ – 1 = 10 (r²– 1)

∴ (r² – 1) (r² + 1) = 10 (r² – 1)

∴ r² + 1 = 10

∴ r² = 10- 1

∴ r² = 9

∴ r = ±3