Question

For a geometric progression, the ratio of sum of the fifth and the third term to the difference of the fifth and the third term is 5:3. Find r.

Answer 1

Here, (T₅ + T₃):(T₅ – T₃) = 5 : 3

Now, T_n = ar^n-1

∴ T₅ = ar⁴ and T₃ = ar²

(T₅ + T₃) : (T₅ – T₃) = 5 :3

∴ \(\frac{ar^4+ar^2}{ar^4−ar^2}=\frac{5}{3}\)

∴ \(\frac{ar^2(r^2+1)}{ar^2(r^2−1)}=\frac{5}{3}\)

∴ \(\frac{r^2+1}{r^2−1}=\frac{5}{3}\)

∴ 3r² + 3 = 5r² – 5

∴ 3 + 5 = 5r² – 3r²

∴ 8 = 2r²

∴ r² = 82 = 4

∴ r = ± 2