Problem: Find the common ratio of a geometric progression (G.P) whose sum of first 5 terms is 16 and the sum of first 10 terms is 3904.

Solution:

First, we need to understand the formula for the sum of a G.P:

S = a(1 - r^n) / (1 - r) where a is the first term, r is the common ratio and n is the number of terms.

Finding the common ratio:

Let us assume the first term of the G.P to be a and the common ratio to be r.

We are given that the sum of the first 5 terms is 16, so we can write:

a(1 - r^5) / (1 - r) = 16 ---(1)

Similarly, the sum of the first 10 terms is 3904, so we can write:

a(1 - r^10) / (1 - r) = 3904 ---(2)

Now, we can divide equation (2) by equation (1) to eliminate 'a' and simplify the equation:

(1 - r^10) / (1 - r^5) = 3904 / 16

Simplifying further, we get:

(1 + r^5) (1 - r^5) / (1 - r^5) = 244

1 + r^5 = 244

r^5 = 243

r = 3

Therefore, the common ratio of the given G.P is 3.

Explanation:

To find the common ratio of a G.P, we need to use the formula for the sum of a G.P and solve for the common ratio. In this problem, we are given the sum of the first 5 and first 10 terms of the G.P. We use these values to form two equations and then eliminate the first term 'a' by dividing one equation by the other. This gives us an equation in terms of the common ratio 'r' which we can solve to get the value of 'r'.