Read here to learn about Geometric progression. In this blog, we are going to learn few approaches of solving Geometric Progression problems with solutions.

## Solved problems on Geometric Progression

**Example 1**:

Find the 20th and nth terms of the G.P. *Solution*:

We have the first two terms of the G.P: a_{1} = 5/2, a_{2} = 5/4.

Common ratio of the G.P, r = a_{n}/a_{n-1 }= a_{2}/a_{1} =

Formula for the nth term of G.P. is, a_{n} = ar^{n-1}, where a = first term, r = common ratio.

Therefore, 20th term

Answer = 5/2^{20}.

**Example 2:**

Evaluate

Solution:

The given summations can be written as two summations.

The first summations adds the number 2 eleven times. So, the value of the summation = 11 x 2 = 22.

The second summation can be expanded as

This is a G.P. with first term, a = 3 and r = a2/a1 = 9/3 = 3.

We need to find the sum of first 11 terms of it. The formula for the sum of first n terms of a G.P. is

Substituting the values of a, r, n in it, gives us :

The result of the second summations is 265719.

Answer = first summation + second summation

= 22 + 265719 **Answer = 265741**

#### More Examples:

**Example 3:**

The sum of first three terms of a G.P. is 39/10 and their product is 1. Find the common ratios and the terms.

Solution: **Note**: When we are given the product of odd number (2n + 1) of terms of a G.P. It is preferred to take the sequence as

. By taking the terms as stated when the product of all of the terms is taken, we are left with a^{2n+1}, which reduces our effort significantly.

In the questions, we are given first three terms of G.P. So we consider the terms as .

We know that the sum of the terms is 39/10 and the product of the terms is 1, ⇒ a^{3} = 1 ⇒ a = 1. (real solution)

Now solving for using the equation of sum, we get:

Substituting the values in the expression we get the solutions as r = 0.4 or r = 2.5.

For r = 0.4 and a = 1, the terms of the G.P. are 1/0.4, 1, 0.4 = **2.5, 1, 0.4** .

For r = 2.5 and a = 1, the terms of the G.P. are 1/2.5, 1, 2.5 = **0.4, 1, 2.5** .

The above three approaches can be used to get solutions for a ton of problems on Geometric Progression.

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