If the `p^(th), q^(th)` and `r^(th)` terms of a GP. Are again in G.P., then which one of the following is correct?

Options

A. p,q ,r are in A.P
B. p.q.r are in GP.
C. p,q,r are in H.P.
D. p,q,r are neither in A.P. Nor in G.P. Nor in H.P.

Answer

To solve the problem, we need to establish the relationship between the terms of a geometric progression (GP) and how they relate to being in another GP. Let's break down the solution step by step.

### Step-by-Step Solution:

**Step 1: Define the terms of the GP.**
Let the first term of the GP be \( a \) and the common ratio be \( r \). The \( n^{th} \) term of the GP can be expressed as:
\[ T_n = a \cdot r^{n-1} \]

**Step 2: Write the expressions for the \( p^{th} \), \( q^{th} \), and \( r^{th} \) terms.**
Using the formula for the \( n^{th} \) term, we can write:
\[ T_p = a \cdot r^{p-1} \]
\[ T_q = a \cdot r^{q-1} \]
\[ T_r = a \cdot r^{r-1} \]

**Step 3: Set up the condition for these terms to be in GP.**
For the terms \( T_p, T_q, T_r \) to be in GP, the following condition must hold:
\[ \frac{T_r}{T_q} = \frac{T_q}{T_p} \]

**Step 4: Substitute the expressions into the condition.**
Substituting the expressions we derived:
\[ \frac{a \cdot r^{r-1}}{a \cdot r^{q-1}} = \frac{a \cdot r^{q-1}}{a \cdot r^{p-1}} \]
This simplifies to:
\[ \frac{r^{r-1}}{r^{q-1}} = \frac{r^{q-1}}{r^{p-1}} \]

**Step 5: Simplify the fractions.**
This can be simplified further:
\[ r^{(r-1) - (q-1)} = r^{(q-1) - (p-1)} \]
Which leads to:
\[ r^{r - q} = r^{q - p} \]

**Step 6: Equate the exponents.**
Since the bases are the same (assuming \( r \neq 0 \)), we can equate the exponents:
\[ r - q = q - p \]

**Step 7: Rearranging the equation.**
Rearranging gives us:
\[ r + p = 2q \]

**Step 8: Conclusion.**
This shows that \( p, q, r \) are in an arithmetic progression (AP) because the middle term \( q \) is the average of \( p \) and \( r \). Therefore, the correct conclusion is:
\[ \text{If the } p^{th}, q^{th}, \text{ and } r^{th} \text{ terms of a GP are in GP, then } p, q, r \text{ are in AP.} \]

### Final Answer:
The correct option is that \( p, q, r \) are in AP.