The sum of first three terms of a G.P. is 21 and sum of their squares is 189. Find the common ratio :

Options

A. `1/2` or 2
B. 3 or `1/3`
C. 4 or `1/4`
D. none of these

Answer

To solve the problem step by step, we will denote the first three terms of the geometric progression (G.P.) as \( a \), \( ar \), and \( ar^2 \), where \( a \) is the first term and \( r \) is the common ratio.

### Step 1: Set up the equations from the problem statement
From the problem, we know:
1. The sum of the first three terms is 21:
\[ a + ar + ar^2 = 21 \]
Factoring out \( a \):
\[ a(1 + r + r^2) = 21 \quad \text{(Equation 1)} \]

2. The sum of their squares is 189:
\[ a^2 + (ar)^2 + (ar^2)^2 = 189 \]
This can be rewritten as:
\[ a^2(1 + r^2 + r^4) = 189 \quad \text{(Equation 2)} \]

### Step 2: Square Equation 1
We will square Equation 1:
\[ [a(1 + r + r^2)]^2 = 21^2 \]
This gives us:
\[ a^2(1 + r + r^2)^2 = 441 \quad \text{(Equation 3)} \]

### Step 3: Divide Equation 3 by Equation 2
Now, we will divide Equation 3 by Equation 2:
\[ \frac{a^2(1 + r + r^2)^2}{a^2(1 + r^2 + r^4)} = \frac{441}{189} \]
The \( a^2 \) cancels out:
\[ \frac{(1 + r + r^2)^2}{1 + r^2 + r^4} = \frac{441}{189} \]
Simplifying \( \frac{441}{189} \) gives:
\[ \frac{441 \div 63}{189 \div 63} = \frac{7}{3} \]
So we have:
\[ \frac{(1 + r + r^2)^2}{1 + r^2 + r^4} = \frac{7}{3} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 3(1 + r + r^2)^2 = 7(1 + r^2 + r^4) \]

### Step 5: Expand both sides
Expanding the left side:
\[ 3(1 + 2r + r^2 + r^2 + 2r^3 + r^4) = 3(1 + 2r + 2r^2 + 2r^3 + r^4) \]
This simplifies to:
\[ 3 + 6r + 6r^2 + 6r^3 + 3r^4 \]

Expanding the right side:
\[ 7(1 + r^2 + r^4) = 7 + 7r^2 + 7r^4 \]

### Step 6: Set the equation to zero
Setting both sides equal:
\[ 3 + 6r + 6r^2 + 6r^3 + 3r^4 = 7 + 7r^2 + 7r^4 \]
Rearranging gives:
\[ 3r^4 - 4r^4 + 6r^3 - 6r + 6r^2 - 7r^2 + 3 - 7 = 0 \]
This simplifies to:
\[ -4r^4 + 6r^3 - r^2 - 6r - 4 = 0 \]
Multiplying through by -1 gives:
\[ 4r^4 - 6r^3 + r^2 + 6r + 4 = 0 \]

### Step 7: Solve the quadratic equation
We can use the quadratic formula or factor to find \( r \). Let's simplify:
\[ 2r^2 - 5r + 2 = 0 \]
Using the quadratic formula:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 2 \), \( b = -5 \), and \( c = 2 \):
\[ r = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} \]
Calculating the discriminant:
\[ r = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm 3}{4} \]
This gives us two potential solutions:
1. \( r = \frac{8}{4} = 2 \)
2. \( r = \frac{2}{4} = \frac{1}{2} \)

### Conclusion
The common ratio \( r \) can be either \( 2 \) or \( \frac{1}{2} \).