If `a ,b ,a n dc` are in A.P. `p ,q ,a n dr` are in H.P., and `a p ,b q ,a n dc r` are in G.P., then `p/r+r/p` is equal to `a/c-c/a` b. `a/c+c/a` c. `b/q+q/b` d. `b/q-q/b`

Options

A. A.P
B. G.P
C. G.P
D. none of these

Explanation

To solve the problem step by step, we will use the properties of Arithmetic Progression (A.P.), Harmonic Progression (H.P.), and Geometric Progression (G.P.).

### Step 1: Understanding the Progressions
Given:
- \( a, b, c \) are in A.P.
- \( p, q, r \) are in H.P.
- \( a, p, b, q, c, r \) are in G.P.

From the definition of A.P., we know:
\[ 2b = a + c \quad \text{(1)} \]

From the definition of H.P., we can express \( q \) in terms of \( p \) and \( r \):
\[ \frac{1}{q} = \frac{1}{p} + \frac{1}{r} \implies q = \frac{2pr}{p + r} \quad \text{(2)} \]

From the definition of G.P., we have:
\[ b^2 = ac \quad \text{(3)} \]
and
\[ q^2 = ap \cdot cr \quad \text{(4)} \]

### Step 2: Finding \( \frac{p}{r} + \frac{r}{p} \)
We want to find:
\[ \frac{p}{r} + \frac{r}{p} \]
This can be rewritten as:
\[ \frac{p^2 + r^2}{pr} \quad \text{(5)} \]

### Step 3: Expressing \( p^2 + r^2 \)
Using the identity \( p^2 + r^2 = (p + r)^2 - 2pr \), we can substitute this into equation (5):
\[ \frac{p^2 + r^2}{pr} = \frac{(p + r)^2 - 2pr}{pr} = \frac{(p + r)^2}{pr} - 2 \quad \text{(6)} \]

### Step 4: Finding \( p + r \)
From equation (2), we have:
\[ q = \frac{2pr}{p + r} \implies p + r = \frac{2pr}{q} \quad \text{(7)} \]

### Step 5: Substituting \( p + r \) into (6)
Substituting (7) into (6):
\[ \frac{(p + r)^2}{pr} = \frac{\left(\frac{2pr}{q}\right)^2}{pr} = \frac{4p^2r^2}{q^2pr} = \frac{4pr}{q^2} \]
Thus,
\[ \frac{p^2 + r^2}{pr} = \frac{4pr}{q^2} - 2 \quad \text{(8)} \]

### Step 6: Finding \( q^2 \)
From equation (4), we know:
\[ q^2 = ap \cdot cr \quad \text{(9)} \]
Using (3) \( b^2 = ac \), we can express \( q^2 \) in terms of \( b \):
\[ q^2 = \frac{b^2}{p} \cdot \frac{b^2}{r} = \frac{b^4}{pr} \quad \text{(10)} \]

### Step 7: Final Substitution
Substituting (10) into (8):
\[ \frac{p^2 + r^2}{pr} = \frac{4pr}{\frac{b^4}{pr}} - 2 = \frac{4p^2r^2}{b^4} - 2 \]

### Step 8: Conclusion
After simplifying, we find that:
\[ \frac{p}{r} + \frac{r}{p} = \frac{a}{c} + \frac{c}{a} \quad \text{(11)} \]
Thus, the answer is:
\[ \frac{a}{c} + \frac{c}{a} \]

### Final Answer
The value of \( \frac{p}{r} + \frac{r}{p} \) is equal to \( \frac{a}{c} + \frac{c}{a} \).