Question

In a Geometric Progression (G.P.) the first term is 24 and the fifth term is 8. Find the ninth term of the G.P.

Answer 1

a = 24 and T₅ = 8, t₉ = ?

t_n = ar^n-1

8 = ar^5-1 

8 = ar⁴

8 = 24r⁴ 

r⁴ = ${\displaystyle \frac{8}{24}}$

r⁴ = ${\displaystyle \frac{1}{3}}$

Taking the fourth root on both sides:

r = ${\displaystyle {\left(\frac{1}{3}\right)}^{\frac{1}{4}}=\frac{1}{\sqrt{3}}}$

∴ t₉ = ?

t₉ = ar^9-1

t₉ = ar⁸

t₉ = ${\displaystyle 24\times {\left(\frac{1}{\sqrt{3}}\right)}^8}$

t₉ = ${\displaystyle 24\times \frac{1}{{\left(\sqrt{3}\right)}^8}}$

t₉ = ${\displaystyle 24\times \frac{1}{81}}$

t₉ = ${\displaystyle \frac{24}{81}}$

t₉ = ${\displaystyle \frac{8}{27}}$

The ninth term of the G.P. is ${\displaystyle \frac{8}{27}}$.