Key Concepts and Formulas

• Geometric Progression (G.P.): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$).
• General Term of a G.P.: If the first term is $a_1$ and the common ratio is $r$, the $n$-th term is given by $a_n = a_1 r^{n-1}$.
• Ratio of Terms in a G.P.: For any two terms $a_m$ and $a_n$ in a G.P., their ratio is $\frac{a_m}{a_n} = r^{m-n}$.

Step-by-Step Solution

Step 1: Express the given ratio in terms of the common ratio. We are given that $a_1, a_2, \dots, a_{10}$ form a Geometric Progression. Let $a_1$ be the first term and $r$ be the common ratio. The $n$-th term of a G.P. is given by $a_n = a_1 r^{n-1}$. We are given the condition $\frac{a_3}{a_1} = 25$. Using the general term formula, $a_3 = a_1 r^{3-1} = a_1 r^2$. Substituting this into the given condition: $\frac{a_1 r^2}{a_1} = 25$ Since $a_1$ is the first term of a G.P., it is non-zero. We can cancel $a_1$: $r^2 = 25$ Why this step? This step directly uses the definition of a G.P. to relate the given numerical ratio to the common ratio of the sequence. It's the crucial first step to find information about $r$.

Step 2: Express the required ratio in terms of the common ratio. We need to find the value of $\frac{a_9}{a_5}$. Using the general term formula: $a_9 = a_1 r^{9-1} = a_1 r^8$ $a_5 = a_1 r^{5-1} = a_1 r^4$ Now, we can write the ratio: $\frac{a_9}{a_5} = \frac{a_1 r^8}{a_1 r^4}$ Again, canceling $a_1$ (since it's non-zero): $\frac{a_9}{a_5} = \frac{r^8}{r^4}$ Using the exponent rule $\frac{x^m}{x^n} = x^{m-n}$: $\frac{a_9}{a_5} = r^{8-4} = r^4$ Why this step? This step transforms the unknown ratio of two specific terms into an expression involving only the common ratio. This is a standard technique in G.P. problems, simplifying the problem to finding a power of $r$.

Step 3: Calculate the value of the required ratio. From Step 1, we found that $r^2 = 25$. From Step 2, we found that $\frac{a_9}{a_5} = r^4$. We can express $r^4$ as $(r^2)^2$. Substituting the value of $r^2$: $\frac{a_9}{a_5} = (r^2)^2 = (25)^2$ Calculating the square of 25: $(25)^2 = 625$ To express this in terms of powers of 5, we know $25 = 5^2$. So, $625 = (25)^2 = (5^2)^2 = 5^{2 \times 2} = 5^4$. Why this step? This is the final calculation. By using the result from Step 1 ($r^2=25$) in the expression derived in Step 2 ($r^4$), we arrive at the numerical value of the required ratio. Expressing it as $5^4$ helps match the given options.

The value of $\frac{a_9}{a_5}$ is $5^4$.

Common Mistakes & Tips

• Incorrect General Term: Ensure you use the correct formula for the $n$-th term of a G.P., which is $a_n = a_1 r^{n-1}$. Using $a_n = a_1 r^n$ will lead to incorrect exponents.
• Calculating 'r' Unnecessarily: In this problem, we only needed $r^2$ and $r^4$. There was no need to find the value of $r$ itself (which could be $5$ or $-5$). Working with $r^2$ directly simplifies the calculation.
• Exponent Manipulation: Be fluent with exponent rules, especially $(x^a)^b = x^{ab}$, which is crucial for simplifying powers of the common ratio.

Summary

The problem involves a Geometric Progression where the ratio of the third term to the first term is given. By expressing the terms using the general formula $a_n = a_1 r^{n-1}$, we find that $\frac{a_3}{a_1} = r^2 = 25$. We then need to find $\frac{a_9}{a_5}$, which simplifies to $r^{9-5} = r^4$. Since $r^4 = (r^2)^2$, we can substitute $r^2 = 25$ to get $(25)^2 = 625$, which is $5^4$.

The final answer is $\boxed{5^4}$.