Let `A_(1),A_(2),A_(3),"......."A_(m)` be arithmetic means between `-3` and 828 and `G_(1),G_(2),G_(3),"......."G_(n)` be geometric means between 1 and 2187. Produmt of geometrimc means is `3^(35)` and sum of arithmetic means is 14025.
The valjue of n is

Options

A. 45
B. 30
C. 25
D. 10

Explanation

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Understanding the Problem
We have two sets of means:
1. Arithmetic means \( A_1, A_2, A_3, \ldots, A_m \) between -3 and 828.
2. Geometric means \( G_1, G_2, G_3, \ldots, G_n \) between 1 and 2187.

We are given:
- The product of the geometric means is \( 3^{35} \).
- The sum of the arithmetic means is 14025.

We need to find the value of \( n \).

### Step 2: Finding the Number of Arithmetic Means
The arithmetic means can be calculated using the formula for the sum of an arithmetic series. The number of arithmetic means \( m \) between two numbers \( a \) and \( b \) can be calculated as follows:

\[ \text{Sum} = \frac{m + 2}{2} \times \text{(first term + last term)} \]

Here, the first term is -3 and the last term is 828. Thus, we have:

\[ 14025 = \frac{m + 2}{2} \times (-3 + 828) \]

Calculating \( -3 + 828 = 825 \):

\[ 14025 = \frac{m + 2}{2} \times 825 \]

### Step 3: Solving for \( m \)
Now, we can rearrange the equation to solve for \( m \):

\[ 14025 = \frac{m + 2}{2} \times 825 \]

Multiplying both sides by 2:

\[ 28050 = (m + 2) \times 825 \]

Now, divide both sides by 825:

\[ m + 2 = \frac{28050}{825} \]

Calculating \( \frac{28050}{825} = 34 \):

\[ m + 2 = 34 \]

Thus, we find:

\[ m = 34 - 2 = 32 \]

### Step 4: Finding the Number of Geometric Means
Next, we use the information about the geometric means. The product of the geometric means is given by:

\[ \text{Product} = G_1 \cdot G_2 \cdot \ldots \cdot G_n = \left( \frac{1 \cdot 2187}{2} \right)^{n} \]

We know that \( 2187 = 3^7 \), so we can express the product as:

\[ 3^{35} = (1 \cdot 2187)^{n/2} = (1 \cdot 3^7)^{n/2} \]

This simplifies to:

\[ 3^{35} = 3^{7n/2} \]

### Step 5: Equating the Exponents
Since the bases are the same, we can equate the exponents:

\[ 35 = \frac{7n}{2} \]

Multiplying both sides by 2:

\[ 70 = 7n \]

Now, dividing both sides by 7:

\[ n = 10 \]

### Final Answer
The value of \( n \) is \( \boxed{10} \).

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