Let a, b and c be the 7th, 11th and 13th terms, respectively, of a non-constant A.P.. If these are also the three consecutive terms of a G.P., then `(a)/(c )` is equal to

Options

A. `1//2`
B. 4
C. 2
D. `7//13`

Answer

To solve the problem step by step, we will first express the terms of the arithmetic progression (A.P.) and then use the property of geometric progression (G.P.) to find the required ratio.

### Step 1: Define the terms of the A.P.
Let the first term of the A.P. be \( a_1 \) and the common difference be \( d \). The \( n \)-th term of an A.P. can be expressed as:
\[ T_n = a_1 + (n-1)d \]
Thus, we can express the 7th, 11th, and 13th terms as follows:
- 7th term \( a = a_1 + 6d \)
- 11th term \( b = a_1 + 10d \)
- 13th term \( c = a_1 + 12d \)

### Step 2: Use the property of G.P.
Since \( a, b, c \) are also in G.P., we can use the property that for three numbers to be in G.P., the square of the middle term must equal the product of the other two terms:
\[ b^2 = a \cdot c \]
Substituting the expressions for \( a, b, c \):
\[ (a_1 + 10d)^2 = (a_1 + 6d)(a_1 + 12d) \]

### Step 3: Expand both sides of the equation
Expanding the left-hand side:
\[ (a_1 + 10d)^2 = a_1^2 + 20a_1d + 100d^2 \]
Expanding the right-hand side:
\[ (a_1 + 6d)(a_1 + 12d) = a_1^2 + 12a_1d + 6a_1d + 72d^2 = a_1^2 + 18a_1d + 72d^2 \]

### Step 4: Set the two expansions equal to each other
Now we set the two expansions equal:
\[ a_1^2 + 20a_1d + 100d^2 = a_1^2 + 18a_1d + 72d^2 \]

### Step 5: Simplify the equation
Subtract \( a_1^2 \) from both sides:
\[ 20a_1d + 100d^2 = 18a_1d + 72d^2 \]
Now, subtract \( 18a_1d \) and \( 72d^2 \) from both sides:
\[ 2a_1d + 28d^2 = 0 \]

### Step 6: Factor out common terms
Factoring out \( d \) (since \( d \neq 0 \)):
\[ 2a_1 + 28d = 0 \]
This simplifies to:
\[ a_1 = -14d \]

### Step 7: Substitute back to find \( a, b, c \)
Now substituting \( a_1 \) back into the expressions for \( a, b, c \):
- \( a = -14d + 6d = -8d \)
- \( b = -14d + 10d = -4d \)
- \( c = -14d + 12d = -2d \)

### Step 8: Find the ratio \( \frac{a}{c} \)
Now we can find the ratio:
\[ \frac{a}{c} = \frac{-8d}{-2d} = \frac{8}{2} = 4 \]

### Final Answer
Thus, the value of \( \frac{a}{c} \) is \( 4 \).