The 5th , 8th and 11th terms of a G.P. are a,b,c respectively , then which one of the following is true ?

Options

A. a)2b=ac
B. b)`b^2 = ac`
C. c)a+b+c=0
D. d)none of these

Answer

To solve the problem, we need to find the relationship between the 5th, 8th, and 11th terms of a geometric progression (G.P.) given as \( a \), \( b \), and \( c \) respectively.

### Step-by-Step Solution:

1. **Understand the nth term of a G.P.**:
The nth term of a geometric progression can be expressed as:
\[ T_n = A \cdot r^{n-1} \]
where \( A \) is the first term and \( r \) is the common ratio.

2. **Write the expressions for the 5th, 8th, and 11th terms**:
- The 5th term \( T_5 \) is:
\[ T_5 = A \cdot r^{4} = a \]
- The 8th term \( T_8 \) is:
\[ T_8 = A \cdot r^{7} = b \]
- The 11th term \( T_{11} \) is:
\[ T_{11} = A \cdot r^{10} = c \]

3. **Express \( A \) in terms of \( a \)**:
From the equation for \( T_5 \):
\[ A = \frac{a}{r^4} \]

4. **Substitute \( A \) into the equations for \( b \) and \( c \)**:
- For \( b \):
\[ b = A \cdot r^{7} = \frac{a}{r^4} \cdot r^{7} = a \cdot r^{3} \]
- For \( c \):
\[ c = A \cdot r^{10} = \frac{a}{r^4} \cdot r^{10} = a \cdot r^{6} \]

5. **Establish a relationship between \( a \), \( b \), and \( c \)**:
We can express \( b \) and \( c \) in terms of \( a \):
- From \( b = a \cdot r^{3} \)
- From \( c = a \cdot r^{6} \)

6. **Find the relationship \( b^2 = ac \)**:
- We can square \( b \):
\[ b^2 = (a \cdot r^{3})^2 = a^2 \cdot r^{6} \]
- Now, calculate \( ac \):
\[ ac = a \cdot (a \cdot r^{6}) = a^2 \cdot r^{6} \]
- Thus, we have:
\[ b^2 = ac \]

### Conclusion:
The relationship that holds true is:
\[ b^2 = ac \]