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

Answer

To solve the problem step by step, let's break it down:

**Step 1: Understand the terms of the G.P.**
- The first term of the G.P. (Geometric Progression) is given as \( A = 1 \).
- The common ratio is denoted as \( r \).
- The terms of the G.P. can be expressed as follows:
- The 3rd term: \( A_3 = A \cdot r^2 = 1 \cdot r^2 = r^2 \)
- The 5th term: \( A_5 = A \cdot r^4 = 1 \cdot r^4 = r^4 \)

**Step 2: Set up the equation based on the given information.**
- We know that the sum of the 3rd and 5th terms is 90:
\[ A_3 + A_5 = 90 \]
Substituting the expressions for \( A_3 \) and \( A_5 \):
\[ r^2 + r^4 = 90 \]

**Step 3: Rearrange the equation.**
- We can rearrange the equation to factor out \( r^2 \):
\[ r^4 + r^2 - 90 = 0 \]
Let \( x = r^2 \). Then the equation becomes:
\[ x^2 + x - 90 = 0 \]

**Step 4: Solve the quadratic equation.**
- We can use the quadratic formula to solve for \( x \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 1 \), and \( c = -90 \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-90)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 360}}{2} \]
\[ x = \frac{-1 \pm \sqrt{361}}{2} \]
\[ x = \frac{-1 \pm 19}{2} \]

**Step 5: Calculate the values of \( x \).**
- We have two potential solutions:
1. \( x = \frac{18}{2} = 9 \)
2. \( x = \frac{-20}{2} = -10 \) (not valid since \( x = r^2 \) cannot be negative)

**Step 6: Find the common ratio \( r \).**
- Since \( x = r^2 \), we have:
\[ r^2 = 9 \]
Therefore, \( r = \sqrt{9} = 3 \) or \( r = -3 \). However, we typically consider the positive common ratio in G.P., so:
\[ r = 3 \]

**Final Answer:**
The common ratio of the G.P. is \( r = 3 \).

---