The first and second terms of a G.P are `x^(-4)` and `x^(m)` respectively. If its 8th term is `x^(52)`, then the value of m is

Options

A. A. 8
B. B. 6
C. C. 4
D. D. 2

Answer

To solve the problem step by step, we start with the information given about the geometric progression (G.P.).

### Step 1: Identify the first and second terms
The first term \( a_1 \) is given as:
\[ a_1 = x^{-4} \]
The second term \( a_2 \) is given as:
\[ a_2 = x^{m} \]

### Step 2: Find the common ratio \( r \)
In a G.P., the second term can be expressed in terms of the first term and the common ratio \( r \):
\[ a_2 = a_1 \cdot r \]
Substituting the known values:
\[ x^{m} = x^{-4} \cdot r \]
From this, we can express \( r \):
\[ r = \frac{x^{m}}{x^{-4}} = x^{m + 4} \]

### Step 3: Use the formula for the 8th term
The 8th term \( a_8 \) of a G.P. is given by:
\[ a_8 = a_1 \cdot r^{7} \]
We know that \( a_8 = x^{52} \), so we can write:
\[ x^{52} = x^{-4} \cdot r^{7} \]

### Step 4: Substitute \( r \) into the equation
Now, substitute \( r = x^{m + 4} \) into the equation:
\[ x^{52} = x^{-4} \cdot (x^{m + 4})^{7} \]
This simplifies to:
\[ x^{52} = x^{-4} \cdot x^{7(m + 4)} \]
Combining the exponents, we have:
\[ x^{52} = x^{-4 + 7(m + 4)} \]

### Step 5: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 52 = -4 + 7(m + 4) \]

### Step 6: Simplify the equation
First, distribute the 7:
\[ 52 = -4 + 7m + 28 \]
Combine like terms:
\[ 52 = 7m + 24 \]

### Step 7: Solve for \( m \)
Now, isolate \( m \):
\[ 52 - 24 = 7m \]
\[ 28 = 7m \]
\[ m = \frac{28}{7} = 4 \]

### Final Answer
The value of \( m \) is:
\[ \boxed{4} \]