Question

The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Answer 1

The common ratio of the geometric progression is 3, given that the first term is 1, and the sum of the third and fifth terms is 90.

Let the first term of the geometric progression (G.P.) be a and the common ratio be r. The terms of the G.P. are given by
`\( a, ar, ar^2, \ldots \)`.

The sum of the third term and fifth term is given by
`\( ar^2 + ar^4 \)`, and it is given that this sum is equal to 90.

`\[ ar^2 + ar^4 = 90 \]`

Since the first term is 1, we have a = 1.

`\[ r^2 + r^4 = 90 \]`

Now, we can factor this equation:

`\[ r^2(r^2 + 1) = 90 \]\[ r^2(r^2 + 1) - 90 = 0 \]`

Now, we can factor this quadratic equation:

`\[ (r^2 - 9)(r^2 + 10) = 0 \]`

This gives two possible solutions for r^2: r^2 = 9 or r^2 = -10.

However, the common ratio r must be positive, so we discard the solution r^2 = -10.

Therefore, r^2 = 9, and taking the positive square root on both sides:

r = 3

So, the common ratio of the geometric progression is r = 3