(Q15).If the geometric progressions 162, 54, 18, ….. and
2
81
 ,
2
27
 ,
2
9
 , ........ have their nth term equal, find the value of n.

Given G.P.: 162, 54, 18, ….. and
2
81
 ,
2
27
 ,
2
9
 , ......

Here a =

r =
a2
a1
 =
=
=
Here a =
r =
a2
a1
 =
=
×
=

Given that nth terms are equal

an = a . rn-1

⇒ × (
1
 )n-1 =
× ()n-1
⇒ n-1 × n-1 = ×

⇒ n-1+n-1 = ×

⇒ 2n-2 = 4 × 4

⇒ 2n-2 = 8 [∵ am . an = am+n]

⇒ n – 2 =

[∵ bases are equal, exponents are also equal]

n = + 2

⇒ n =
=

The th terms of the two G.P.s are equal.