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 value of m is

Options

A. 17
B. 34
C. 51
D. 68

Explanation

To solve the problem step by step, we will analyze the information given and apply the relevant formulas for arithmetic and geometric means.

### Step 1: Understanding the Arithmetic Means
We have \( A_1, A_2, A_3, \ldots, A_m \) as arithmetic means between -3 and 828. The total number of terms in this arithmetic sequence is \( m + 2 \) (including the first term and the last term).

### Step 2: Sum of Arithmetic Means
The sum of the arithmetic means is given as 14025. The formula for the sum of an arithmetic series is:
\[ S = \frac{n}{2} \times (a + l) \]
where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. Here, \( n = m + 2 \), \( a = -3 \), and \( l = 828 \).

Thus, we can write:
\[ 14025 = \frac{m + 2}{2} \times (-3 + 828) \]

### Step 3: Simplifying the Equation
Calculating \( -3 + 828 \):
\[ -3 + 828 = 825 \]
Now substituting this back into the equation:
\[ 14025 = \frac{m + 2}{2} \times 825 \]

### Step 4: Solving for \( m \)
Multiply both sides by 2 to eliminate the fraction:
\[ 28050 = (m + 2) \times 825 \]
Now, divide both sides by 825:
\[ m + 2 = \frac{28050}{825} \]

Calculating \( \frac{28050}{825} \):
\[ \frac{28050}{825} = 34 \]
So, we have:
\[ m + 2 = 34 \]
Subtracting 2 from both sides gives:
\[ m = 34 - 2 = 32 \]

### Final Answer
Thus, the value of \( m \) is 32.