Let `a_1,a_2 ,a_3` …..be in A.P. and `a_p, a_q , a_r` be in G.P. Then `a_q : a_p` is equal to :

Options

A. `(r-p)/(q-p)`
B. `(q-p)/(r-q)`
C. `(r-q)/(q-p)`
D. `1`

Answer

To solve the problem, we need to find the ratio \( \frac{a_q}{a_p} \) given that \( a_1, a_2, a_3, \ldots \) are in Arithmetic Progression (A.P.) and \( a_p, a_q, a_r \) are in Geometric Progression (G.P.).

### Step-by-Step Solution:

1. **Define the terms in A.P.**:
The general term of an A.P. can be expressed as:
\[ a_n = a_1 + (n-1)d \]
where \( a_1 \) is the first term and \( d \) is the common difference.

2. **Express \( a_p, a_q, a_r \) in terms of \( a_1 \) and \( d \)**:
- \( a_p = a_1 + (p-1)d \)
- \( a_q = a_1 + (q-1)d \)
- \( a_r = a_1 + (r-1)d \)

3. **Using the property of G.P.**:
Since \( a_p, a_q, a_r \) are in G.P., we have:
\[ \frac{a_q}{a_p} = \frac{a_r}{a_q} \]
Let’s denote this common ratio as \( R \):
\[ R = \frac{a_q}{a_p} = \frac{a_r}{a_q} \]

4. **Substituting the expressions**:
Substitute \( a_p, a_q, a_r \) into the ratio:
\[ R = \frac{a_1 + (q-1)d}{a_1 + (p-1)d} \]
and
\[ R = \frac{a_1 + (r-1)d}{a_1 + (q-1)d} \]

5. **Setting up the equation**:
From the equality of the ratios, we can write:
\[ \frac{a_1 + (q-1)d}{a_1 + (p-1)d} = \frac{a_1 + (r-1)d}{a_1 + (q-1)d} \]

6. **Cross-multiplying**:
Cross-multiplying gives us:
\[ (a_1 + (q-1)d)^2 = (a_1 + (p-1)d)(a_1 + (r-1)d) \]

7. **Expanding both sides**:
Expanding the left side:
\[ (a_1 + (q-1)d)(a_1 + (q-1)d) = a_1^2 + 2a_1(q-1)d + (q-1)^2d^2 \]
Expanding the right side:
\[ (a_1 + (p-1)d)(a_1 + (r-1)d) = a_1^2 + (p+r-2)a_1d + (p-1)(r-1)d^2 \]

8. **Setting the expanded forms equal**:
Set the two expanded forms equal and simplify to find a relationship between \( p, q, r \).

9. **Finding \( \frac{a_q}{a_p} \)**:
After simplification, we find:
\[ R = \frac{q - r}{p - q} \]
To express it in a more standard form, we can factor out a negative sign:
\[ R = \frac{r - q}{q - p} \]

### Final Result:
Thus, the ratio \( \frac{a_q}{a_p} \) is given by:
\[ \frac{a_q}{a_p} = \frac{r - q}{q - p} \]