Key Concepts and Formulas

1. Arithmetic Progression (A.P.): For an A.P. with first term $A$ and common difference $d$, the $n^{th}$ term is given by $T_n = A + (n-1)d$. For a non-constant A.P., $d \neq 0$.
2. Geometric Progression (G.P.): If three terms $x, y, z$ are consecutive terms of a G.P., then the middle term is the geometric mean of the other two, i.e., $y^2 = xz$.

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Step-by-Step Solution

Step 1: Express the given terms of the A.P.

Let the first term of the non-constant A.P. be $A$ and its common difference be $d$. We are given that $a$, $b$, and $c$ are the $7^{th}$, $11^{th}$, and $13^{th}$ terms of this A.P., respectively. Using the formula $T_n = A + (n-1)d$:

• The $7^{th}$ term is $a$: $a = A + (7-1)d = A + 6d$
• The $11^{th}$ term is $b$: $b = A + (11-1)d = A + 10d$
• The $13^{th}$ term is $c$: $c = A + (13-1)d = A + 12d$ Since the A.P. is non-constant, $d \neq 0$.

Step 2: Apply the G.P. condition to $a, b, c$.

We are given that $a, b, c$ are three consecutive terms of a G.P. The property of consecutive terms in a G.P. is that the square of the middle term equals the product of the other two. Therefore, we have: $b^2 = ac$ Substitute the expressions for $a, b, c$ from Step 1 into this equation: $(A + 10d)^2 = (A + 6d)(A + 12d)$

Step 3: Solve the equation to find the relationship between $A$ and $d$.

Expand both sides of the equation from Step 2: Left side: $(A + 10d)^2 = A^2 + 2(A)(10d) + (10d)^2 = A^2 + 20Ad + 100d^2$ Right side: $(A + 6d)(A + 12d) = A(A) + A(12d) + (6d)A + (6d)(12d) = A^2 + 12Ad + 6Ad + 72d^2 = A^2 + 18Ad + 72d^2$ Now, equate the expanded forms: $A^2 + 20Ad + 100d^2 = A^2 + 18Ad + 72d^2$ Subtract $A^2$ from both sides: $20Ad + 100d^2 = 18Ad + 72d^2$ Rearrange the terms to one side: $20Ad - 18Ad + 100d^2 - 72d^2 = 0$ $2Ad + 28d^2 = 0$ Factor out $2d$: $2d(A + 14d) = 0$ Since the A.P. is non-constant, $d \neq 0$. Therefore, we can divide by $2d$, which implies: $A + 14d = 0$ This gives us the relationship $A = -14d$.

Step 4: Calculate the required ratio $\frac{a}{c}$.

We need to find the value of $\frac{a}{c}$. Using the expressions from Step 1: $a = A + 6d$ $c = A + 12d$ The ratio is: $\frac{a}{c} = \frac{A + 6d}{A + 12d}$ Substitute the relationship $A = -14d$ into this ratio: $\frac{a}{c} = \frac{(-14d) + 6d}{(-14d) + 12d}$ $\frac{a}{c} = \frac{-8d}{-2d}$ Since $d \neq 0$, we can cancel $d$ from the numerator and denominator: $\frac{a}{c} = \frac{-8}{-2}$ $\frac{a}{c} = 4$

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Common Mistakes & Tips

1. Algebraic Errors: Be meticulous when expanding squares and products and when simplifying equations. Errors in these steps are common.
2. Non-Constant A.P. Condition: Remember that $d \neq 0$ is crucial. This allows you to divide by $d$ when solving equations like $2d(A + 14d) = 0$.
3. Substitution Strategy: After finding a relationship between $A$ and $d$ (e.g., $A = -14d$), substitute it into the expression you need to evaluate ($\frac{a}{c}$) as early as possible to simplify the problem.

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Summary

This problem requires the application of formulas for the $n^{th}$ term of an Arithmetic Progression and the property of consecutive terms in a Geometric Progression. By expressing the given terms $a, b, c$ in terms of the first term $A$ and common difference $d$ of the A.P., and then using the condition $b^2 = ac$ for the G.P., we derived a linear relationship between $A$ and $d$. Substituting this relationship back into the expression for $\frac{a}{c}$ allowed us to find the final numerical value. The condition that the A.P. is non-constant was essential for a unique solution.

The final answer is $\boxed{4}$.