If p,q and r are in A.P then which of the following is / are true ?

Options

A. pth,qth and rth terms of A.P are in A.P
B. pth,qth,and rht terms of G.P are in G.P
C. pth , qth , and rht terms of H.P are in H.P
D. none of these

Explanation

To determine which statements are true if \( p, q, r \) are in Arithmetic Progression (A.P.), we start by recalling the definition of A.P. and the properties associated with it.

### Step-by-Step Solution:

1. **Understanding A.P.**:
If \( p, q, r \) are in A.P., it means that the difference between consecutive terms is constant. This can be expressed mathematically as:
\[ 2q = p + r \]

2. **Equidistant Terms**:
Since \( p, q, r \) are in A.P., we can infer that \( q \) is the average of \( p \) and \( r \). This property can be extended to any terms derived from \( p, q, r \).

3. **Terms in A.P.**:
If we consider the \( p \)-th, \( q \)-th, and \( r \)-th terms of any A.P., they will also be in A.P. This is because the property of having a constant difference holds for any terms in an A.P.

4. **Terms in GP**:
If \( p, q, r \) are in A.P., we can also check if they can be in a Geometric Progression (G.P.). For \( p, q, r \) to be in G.P., the condition \( q^2 = pr \) must hold. However, this is not necessarily true for all values of \( p, q, r \) in A.P. Therefore, the statement regarding G.P. is not always true.

5. **Terms in HP**:
Similarly, for \( p, q, r \) to be in Harmonic Progression (H.P.), the reciprocals \( \frac{1}{p}, \frac{1}{q}, \frac{1}{r} \) must be in A.P. This is true if \( p, q, r \) are in A.P. Hence, this statement is true.

### Conclusion:
Based on the analysis:
- The statement that the \( p \)-th, \( q \)-th, and \( r \)-th terms of A.P. are in A.P. is **true**.
- The statement that the \( p \)-th, \( q \)-th, and \( r \)-th terms of G.P. are in G.P. is **not necessarily true**.
- The statement that the \( p \)-th, \( q \)-th, and \( r \)-th terms of H.P. are in H.P. is **true**.

Thus, the true statements are those regarding A.P. and H.P.

### Summary of True Statements:
- Statement A: True (A.P.)
- Statement B: False (G.P.)
- Statement C: True (H.P.)