Let `A_1,A_2,A_3,….,A_m` be the arithmetic means between -2 and 1027 and `G_1,G_2,G_3,…., G_n` be the gemetric means between 1 and 1024 .The product of gerometric means is `2^(45)` and sum of arithmetic means is `1024 xx 171`
The value of `n

Answer

To solve the problem, we need to find the value of \( n \), which represents the number of geometric means between 1 and 1024. We are given that the product of the geometric means is \( 2^{45} \) and the sum of the arithmetic means between -2 and 1027 is \( 1024 \times 171 \).

### Step 1: Find the number of arithmetic means \( m \)

The arithmetic means \( A_1, A_2, \ldots, A_m \) are the numbers that lie between -2 and 1027. The total number of terms in this arithmetic progression (AP) is \( m + 2 \) (including -2 and 1027).

The sum of an arithmetic series can be calculated using the formula:
\[ S = \frac{n}{2} \times (a + l) \]
where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term.

In our case:
- First term \( a = -2 \)
- Last term \( l = 1027 \)
- Number of terms \( n = m + 2 \)

The sum of the arithmetic means is given as:
\[ S = 1024 \times 171 \]

Substituting into the formula:
\[ 1024 \times 171 = \frac{m + 2}{2} \times (-2 + 1027) \]

Calculating \( -2 + 1027 \):
\[ -2 + 1027 = 1025 \]

Now substituting this back into the equation:
\[ 1024 \times 171 = \frac{m + 2}{2} \times 1025 \]

### Step 2: Solve for \( m \)

Multiplying both sides by 2 to eliminate the fraction:
\[ 2048 \times 171 = (m + 2) \times 1025 \]

Calculating \( 2048 \times 171 \):
\[ 2048 \times 171 = 350,848 \]

Now we have:
\[ 350,848 = (m + 2) \times 1025 \]

Dividing both sides by 1025:
\[ m + 2 = \frac{350848}{1025} = 342 \]

Subtracting 2 from both sides gives:
\[ m = 340 \]

### Step 3: Find the number of geometric means \( n \)

The geometric means \( G_1, G_2, \ldots, G_n \) are the numbers that lie between 1 and 1024. The product of the geometric means is given as \( 2^{45} \).

The product of \( n \) geometric means can be expressed as:
\[ G_1 \times G_2 \times \ldots \times G_n = (1 \times 1024)^{\frac{n}{2}} = 1024^{\frac{n}{2}} \]

Since \( 1024 = 2^{10} \), we can rewrite this as:
\[ 1024^{\frac{n}{2}} = (2^{10})^{\frac{n}{2}} = 2^{5n} \]

Setting this equal to the given product of the geometric means:
\[ 2^{5n} = 2^{45} \]

### Step 4: Solve for \( n \)

Since the bases are the same, we can equate the exponents:
\[ 5n = 45 \]

Dividing both sides by 5 gives:
\[ n = 9 \]

### Final Answer
The value of \( n \) is \( 9 \).