Question

If the first term of a G.P. is 1 and the sum of 3^rd and 5^th terms is 90, then the positive common ratio of the G.P. is

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Answer 1

The Correct Option is C

Given:

• The first term of the geometric progression (G.P.) is 1.
• The sum of the 3rd and 5th terms is 90.

Formula for nth term of a G.P.:

The nth term of a geometric progression is given by:

T_n = a * r^n-1

where a is the first term and r is the common ratio.

Step 1: Express the 3rd and 5th terms of the G.P.

We know the first term (a) is 1, so:

3rd term (T₃) = 1 * r^3-1 = r²

5th term (T₅) = 1 * r^5-1 = r⁴

Step 2: Use the given sum of the 3rd and 5th terms.

The sum of the 3rd and 5th terms is 90:

r² + r⁴ = 90

Step 3: Solve for r.

Factor out r² from the left side:

r²(1 + r²) = 90

Now, let x = r². The equation becomes:

x(1 + x) = 90

Expanding:

x + x² = 90

x² + x - 90 = 0

This is a quadratic equation. We can solve it using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

For the equation x² + x - 90 = 0, a = 1, b = 1, and c = -90. Substituting these values into the quadratic formula:

x = (-1 ± √(1² - 4(1)(-90))) / 2(1)

x = (-1 ± √(1 + 360)) / 2

x = (-1 ± √361) / 2

x = (-1 ± 19) / 2

x = 9 or x = -10

Step 4: Find r.

Since x = r², we have:

r² = 9, so r = 3 (positive root).

Answer:

The positive common ratio of the G.P. is 3.