Key Concepts and Formulas

1. Arithmetic Progression (A.P.): An A.P. is a sequence where the difference between consecutive terms is constant.

   • The $n^{th}$ term of an A.P. with first term $a$ and common difference $d$ is given by $T_n = a + (n-1)d$.
   • A "non-constant A.P." implies $d \neq 0$.

2. Geometric Progression (G.P.): A G.P. is a sequence where the ratio between consecutive terms is constant (the common ratio, $r$).

   • If three terms $A, B, C$ are in G.P., then $B^2 = AC$.
   • The common ratio is $r = \frac{B}{A} = \frac{C}{B}$.

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Step-by-Step Solution

Step 1: Express the specified terms of the A.P. in terms of its first term and common difference.

Let the first term of the non-constant A.P. be $a$ and its common difference be $d$. Since the A.P. is non-constant, we know that $d \neq 0$. Using the formula $T_n = a + (n-1)d$:

• The $2^{nd}$ term is $T_2 = a + (2-1)d = a + d$.
• The $5^{th}$ term is $T_5 = a + (5-1)d = a + 4d$.
• The $9^{th}$ term is $T_9 = a + (9-1)d = a + 8d$.

Explanation: We are given that specific terms of an A.P. are in G.P. To work with this condition algebraically, we first represent these terms using the standard formula for the $n^{th}$ term of an A.P.

Step 2: Apply the condition that these terms form a G.P.

We are given that the $2^{nd}$, $5^{th}$, and $9^{th}$ terms of the A.P. are in G.P. Let these terms be $A = T_2$, $B = T_5$, and $C = T_9$. For these terms to be in G.P., the property $B^2 = AC$ must hold. Substituting the expressions from Step 1: $(a + 4d)^2 = (a + d)(a + 8d)$

Explanation: The definition of a G.P. implies a specific relationship between any three consecutive terms. By setting the square of the middle term equal to the product of the other two, we transform the G.P. condition into an algebraic equation involving $a$ and $d$.

Step 3: Expand and simplify the equation to find a relationship between $a$ and $d$.

Expand both sides of the equation from Step 2: Left side: $(a + 4d)^2 = a^2 + 2(a)(4d) + (4d)^2 = a^2 + 8ad + 16d^2$. Right side: $(a + d)(a + 8d) = a(a + 8d) + d(a + 8d) = a^2 + 8ad + ad + 8d^2 = a^2 + 9ad + 8d^2$.

Equating the expanded sides: $a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2$

Now, simplify the equation by canceling terms and rearranging: Subtract $a^2$ from both sides: $8ad + 16d^2 = 9ad + 8d^2$ Subtract $8ad$ from both sides: $16d^2 = ad + 8d^2$ Subtract $8d^2$ from both sides: $8d^2 = ad$

Since the A.P. is non-constant, $d \neq 0$. We can divide both sides by $d$: $\frac{8d^2}{d} = \frac{ad}{d}$ $8d = a$

Explanation: This step involves standard algebraic manipulation. Expanding the squared term and the product of binomials, then carefully simplifying by cancelling like terms, leads to a linear relationship between $a$ and $d$. The crucial point here is the ability to divide by $d$, which is guaranteed by the problem statement specifying a "non-constant A.P."

Step 4: Calculate the common ratio of the G.P.

The common ratio ($r$) of the G.P. formed by $T_2, T_5, T_9$ can be found by taking the ratio of any term to its preceding term. Let's use $r = \frac{T_5}{T_2}$. $r = \frac{a + 4d}{a + d}$ Substitute the relationship $a = 8d$ found in Step 3 into this expression: $r = \frac{(8d) + 4d}{(8d) + d}$ $r = \frac{12d}{9d}$ Since $d \neq 0$, we can cancel $d$: $r = \frac{12}{9}$ Simplify the fraction: $r = \frac{4}{3}$

Explanation: With the relationship between $a$ and $d$ established, we can now directly compute the common ratio of the G.P. By substituting $a=8d$, the expression for the ratio becomes a numerical value after canceling $d$, providing the answer.

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Common Mistakes & Tips

• Forgetting "non-constant": If the A.P. were constant ($d=0$), all terms would be equal ($a, a, a, \dots$), forming a G.P. with a common ratio of 1. The "non-constant" condition is vital to exclude this case and allow division by $d$.
• Algebraic Errors: Be meticulous when expanding $(a+4d)^2$ and $(a+d)(a+8d)$ to avoid sign errors or missing terms.
• Confusing A.P. and G.P. parameters: Keep the common difference ($d$) of the A.P. distinct from the common ratio ($r$) of the G.P.

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Summary

The problem requires combining concepts from both arithmetic and geometric progressions. We began by expressing the $2^{nd}$, $5^{th}$, and $9^{th}$ terms of the A.P. in terms of its first term ($a$) and common difference ($d$). The condition that these terms form a G.P. led to an equation ($(a+4d)^2 = (a+d)(a+8d)$), which upon simplification yielded a linear relationship between $a$ and $d$ ($a=8d$). Finally, this relationship was substituted into the formula for the common ratio of the G.P. ($r = \frac{a+4d}{a+d}$), resulting in the common ratio being $\frac{4}{3}$. The "non-constant" A.P. condition was essential for dividing by $d$.

The final answer is $\boxed{\text{D}}$.