Let `a_1,a_2," ..."a_10` be a G.P. If `a_3/a_1=25," then "a_9/a_5` equals

Options

A. `2(5^2)`
B. `4(5^2)`
C. `5^4`
D. `5^3`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the General Terms of a Geometric Progression (G.P.)
In a G.P., the general term \( a_n \) can be expressed as:
\[ a_n = a \cdot r^{n-1} \]
where \( a \) is the first term and \( r \) is the common ratio.

### Step 2: Express \( a_3 \) and \( a_1 \)
From the formula for the general term, we can express:
- \( a_3 = a \cdot r^{3-1} = a \cdot r^2 \)
- \( a_1 = a \)

### Step 3: Set Up the Given Condition
We know from the problem statement that:
\[ \frac{a_3}{a_1} = 25 \]
Substituting the expressions from Step 2:
\[ \frac{a \cdot r^2}{a} = 25 \]
This simplifies to:
\[ r^2 = 25 \]

### Step 4: Find the Value of \( r \)
Taking the square root of both sides, we find:
\[ r = 5 \quad \text{or} \quad r = -5 \]
(However, since we are looking for a ratio, we will consider \( r = 5 \) for simplicity.)

### Step 5: Express \( a_9 \) and \( a_5 \)
Now, we need to find \( \frac{a_9}{a_5} \):
- \( a_9 = a \cdot r^{9-1} = a \cdot r^8 \)
- \( a_5 = a \cdot r^{5-1} = a \cdot r^4 \)

### Step 6: Set Up the Ratio \( \frac{a_9}{a_5} \)
Now we can express the ratio:
\[ \frac{a_9}{a_5} = \frac{a \cdot r^8}{a \cdot r^4} \]
This simplifies to:
\[ \frac{a_9}{a_5} = \frac{r^8}{r^4} = r^{8-4} = r^4 \]

### Step 7: Substitute the Value of \( r \)
Now substituting \( r = 5 \):
\[ \frac{a_9}{a_5} = 5^4 \]

### Step 8: Calculate \( 5^4 \)
Calculating \( 5^4 \):
\[ 5^4 = 625 \]

### Final Answer
Thus, the value of \( \frac{a_9}{a_5} \) is:
\[ \frac{a_9}{a_5} = 625 \]

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