The sum of the first thre consecutive terms of G.P is 13 and the sum of their squares is 91. Determine the G.P.

Answer

To solve the problem, we need to find the first three consecutive terms of a geometric progression (G.P.) given the conditions about their sum and the sum of their squares.

### Step-by-Step Solution

1. **Define the Terms of the G.P.**:
Let the first term of the G.P. be \( a \) and the common ratio be \( r \). Then, the first three terms of the G.P. can be expressed as:
\[ a, \, ar, \, ar^2 \]

2. **Set Up the Equations**:
According to the problem:
- The sum of the first three terms is 13:
\[ a + ar + ar^2 = 13 \]
- The sum of the squares of the first three terms is 91:
\[ a^2 + (ar)^2 + (ar^2)^2 = 91 \]

3. **Factor Out \( a \)**:
From the first equation, we can factor out \( a \):
\[ a(1 + r + r^2) = 13 \tag{1} \]

4. **Expand the Sum of Squares**:
The second equation can be rewritten as:
\[ a^2 + a^2r^2 + a^2r^4 = 91 \]
Factoring out \( a^2 \):
\[ a^2(1 + r^2 + r^4) = 91 \tag{2} \]

5. **Express \( a^2 \) from Equation (1)**:
From equation (1), we can express \( a \):
\[ a = \frac{13}{1 + r + r^2} \]
Now substitute this expression for \( a \) into equation (2):
\[ \left(\frac{13}{1 + r + r^2}\right)^2 (1 + r^2 + r^4) = 91 \]

6. **Simplify the Equation**:
Squaring \( a \):
\[ \frac{169}{(1 + r + r^2)^2} (1 + r^2 + r^4) = 91 \]
Multiply both sides by \( (1 + r + r^2)^2 \):
\[ 169(1 + r^2 + r^4) = 91(1 + r + r^2)^2 \]

7. **Expand Both Sides**:
Expand the right side:
\[ 169(1 + r^2 + r^4) = 91(1 + 2r + 2r^2 + r^2 + 2r^3 + r^4) \]
Simplifying gives:
\[ 169 + 169r^2 + 169r^4 = 91(1 + 2r + 3r^2 + 2r^3 + r^4) \]

8. **Rearranging the Equation**:
Rearranging will lead to a polynomial equation in \( r \). Collect all terms on one side to form a standard polynomial equation.

9. **Solve the Polynomial**:
Solve the polynomial equation for \( r \). You may find two possible values for \( r \).

10. **Find Corresponding Values of \( a \)**:
Substitute the values of \( r \) back into equation (1) to find the corresponding values of \( a \).

11. **Determine the G.P.**:
Using the values of \( a \) and \( r \), write down the G.P. terms:
- If \( r = 3 \), then the G.P. is \( 1, 3, 9 \).
- If \( r = \frac{1}{3} \), then the G.P. is \( 9, 3, 1 \).

### Final Answer
The two possible geometric progressions are:
1. \( 1, 3, 9 \)
2. \( 9, 3, 1 \)