Solved Examples on Geometric Progression

Illustration 1: An infinite G.P. has its first term as x and sum as 5. Then what is the range of x? (2004)

Solution: Solution: we know that the sum of an infinite G.P. is

S_∞ = a/ (1-r), if |r| < 1

      = ∞,            if |r| ≥ 1

Hence, S_∞ = x/(1-r) = 5

Or 1-r = x/5

Hence, r = (5-x)/5 exists only when |r| < 1

Hence, -1 < (5-x)/5 <1

-10 < -x < 0

So this gives 0 < x < 10.

Illustration 2:  The third term of a geometric progression is 4. What is the product of the first five terms?

Solution: here it is given that t₃ = 4.

Hence, this means ar² =4

Now product of first five terms =

a.ar.ar².ar³.ar⁴

= a⁵r¹⁰

= (ar²)⁵

= 4⁵

Illustration: Consider an infinite geometric series with first term ‘a’ and common ratio ‘r’. Find the values of ‘a’ and ‘r’ if its second term is ¾ and its sum is 4.

Solution: It is given in the question that the second term is ¾ and the sum is 4.

Further, the first term is ‘a’ and the common ratio is ‘r’.

Hence, we have a/ (1-r) = 4 and ar = 3/4.

This gives the value of r as 3/4a.

So, 4a²/(4a-3) = 4

This gives (a-1)(a-3) = 0

Hence, a = 1 or 3.

When a = 1 then r =3/4 and when a = 3 then r = 1/4.