Question

Answer the following:

If for a G.P. t₃ = ${\displaystyle \frac{1}{3}}$, t₆ = ${\displaystyle \frac{1}{81}}$ find r

Answer 1

Given, t₃ = ${\displaystyle \frac{1}{3}}$, t₆ = ${\displaystyle \frac{1}{81}}$

t_n = ar^n–1

∴ t₃ = ar² 

∴ ar² = ${\displaystyle \frac{1}{3}}$

∴ a = ${\displaystyle \frac{1}{3{\text{r}}^2}}$    ...(i)

Also, t₆ = ar⁵

∴ ar⁵ = ${\displaystyle \frac{1}{81}}$

∴ ${\displaystyle \frac{1}{3{\text{r}}^2}\times {\text{r}}^5=\frac{1}{81}}$   ...[From (i)]

∴ r³ = ${\displaystyle \frac{1}{27}}$

∴ r = ${\displaystyle \frac{1}{3}}$