Let a,b,c be in A.P. If p is the A.M. between a and b and q is the A.M between b and c, then b is equal to

Options

A. A. `(p+q)/2`
B. B. `(p-q)/2`
C. C. `(pq)/2`
D. D. None of these

Answer

To solve the problem, we need to find the value of \( b \) given that \( a, b, c \) are in Arithmetic Progression (A.P.), and \( p \) and \( q \) are the Arithmetic Means (A.M.) between \( a \) and \( b \), and between \( b \) and \( c \) respectively.

### Step-by-Step Solution:

1. **Understanding A.P.**:
Since \( a, b, c \) are in A.P., we know that:
\[ 2b = a + c \quad \text{(Equation 1)} \]

2. **Finding A.M. between \( a \) and \( b \)**:
The A.M. \( p \) between \( a \) and \( b \) can be expressed as:
\[ p = \frac{a + b}{2} \]
Multiplying both sides by 2 gives:
\[ a + b = 2p \quad \text{(Equation 2)} \]

3. **Finding A.M. between \( b \) and \( c \)**:
The A.M. \( q \) between \( b \) and \( c \) can be expressed as:
\[ q = \frac{b + c}{2} \]
Multiplying both sides by 2 gives:
\[ b + c = 2q \quad \text{(Equation 3)} \]

4. **Adding Equations**:
Now, we will add Equation 2 and Equation 3:
\[ (a + b) + (b + c) = 2p + 2q \]
This simplifies to:
\[ a + 2b + c = 2p + 2q \]

5. **Substituting Equation 1**:
From Equation 1, we know \( a + c = 2b \). We can substitute this into the equation:
\[ 2b + 2b = 2p + 2q \]
This simplifies to:
\[ 4b = 2p + 2q \]

6. **Solving for \( b \)**:
Dividing both sides by 4 gives:
\[ b = \frac{p + q}{2} \]

### Final Answer:
Thus, the value of \( b \) is:
\[ b = \frac{p + q}{2} \]