In a GP of real number , the sum of first four number is 30 and sum of their squares is 340 . The third term of the series is.

Options

A. a. 5
B. b. 16
C. c. 8
D. d. 10

Answer

To solve the problem, we need to find the third term of a geometric progression (GP) where the sum of the first four terms is 30 and the sum of their squares is 340.

### Step-by-step Solution:

1. **Define the terms of the GP**:
Let the first term be \( a \) and the common ratio be \( r \). The first four terms of the GP can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)

2. **Set up the equations**:
From the problem statement, we have two equations:
- The sum of the first four terms:
\[ a + ar + ar^2 + ar^3 = 30 \]
This can be factored as:
\[ a(1 + r + r^2 + r^3) = 30 \quad \text{(Equation 1)} \]

- The sum of the squares of the first four terms:
\[ a^2 + (ar)^2 + (ar^2)^2 + (ar^3)^2 = 340 \]
This can be simplified to:
\[ a^2(1 + r^2 + r^4 + r^6) = 340 \quad \text{(Equation 2)} \]

3. **Express the sums in terms of \( a \) and \( r \)**:
From Equation 1, we can express \( a \) in terms of \( r \):
\[ a = \frac{30}{1 + r + r^2 + r^3} \]

4. **Substitute \( a \) into Equation 2**:
Substitute \( a \) into Equation 2:
\[ \left(\frac{30}{1 + r + r^2 + r^3}\right)^2(1 + r^2 + r^4 + r^6) = 340 \]
Simplifying this gives:
\[ \frac{900(1 + r^2 + r^4 + r^6)}{(1 + r + r^2 + r^3)^2} = 340 \]
Cross-multiplying yields:
\[ 900(1 + r^2 + r^4 + r^6) = 340(1 + r + r^2 + r^3)^2 \]

5. **Simplify and solve for \( r \)**:
After simplifying, we can derive a polynomial equation in terms of \( r \). Solving this polynomial will give us the values of \( r \).

6. **Find \( a \)**:
Once we have the value of \( r \), we can substitute back to find \( a \) using Equation 1.

7. **Calculate the third term**:
The third term of the GP is given by:
\[ ar^2 \]
Substitute the values of \( a \) and \( r \) to find the third term.

### Final Calculation:
After solving the polynomial, we find:
- \( r = 2 \)
- \( a = 2 \)

Thus, the third term is:
\[ ar^2 = 2 \cdot 2^2 = 2 \cdot 4 = 8 \]

### Conclusion:
The third term of the series is **8**.

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