If the geometric progressions 162, 54, 18 …. and have their nth term equal. Find the value of n.

Given series 162, 54, 18 ….

First term (a₁) = 162

Common ratio (r₁) = = =

, …..

First term (a₂) =

Common ratio (r₂) = = = 3

n^th term of both the series is equal.

a₁ r₁^n-1 = a₂r₂^n-1

162×()^n-1 = × 3^n-1

81² = 3^2n-2

3⁸ = 3^2n-2

Comparing the powers on both sides

8 = 2n-2

10 = 2n

n = 5