The first and second terms of a G.P. are `x^(-4) and x^(n)` respectively. If `x^(52)` is the 8th term of the same progression, then n is equal to

Options

A. 13
B. 4
C. 5
D. 3

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Identify the first term and the second term of the G.P.
The first term \( a \) is given as:
\[ a = x^{-4} \]
The second term is given as:
\[ a_2 = x^n \]

### Step 2: Find the common ratio \( r \)
The common ratio \( r \) of a geometric progression (G.P.) is defined as the ratio of the second term to the first term:
\[ r = \frac{a_2}{a} = \frac{x^n}{x^{-4}} = x^{n + 4} \]

### Step 3: Write the formula for the 8th term of the G.P.
The formula for the \( n \)-th term of a G.P. is given by:
\[ T_n = a \cdot r^{n-1} \]
For the 8th term (\( n = 8 \)):
\[ T_8 = a \cdot r^{8-1} = a \cdot r^7 \]
Substituting the values of \( a \) and \( r \):
\[ T_8 = x^{-4} \cdot (x^{n + 4})^7 \]
This simplifies to:
\[ T_8 = x^{-4} \cdot x^{7(n + 4)} = x^{-4 + 7(n + 4)} \]

### Step 4: Set the expression for the 8th term equal to \( x^{52} \)
We know that the 8th term is also given as \( x^{52} \):
\[ x^{-4 + 7(n + 4)} = x^{52} \]

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ -4 + 7(n + 4) = 52 \]

### Step 6: Simplify the equation
Expanding the left side:
\[ -4 + 7n + 28 = 52 \]
Combining like terms:
\[ 7n + 24 = 52 \]

### Step 7: Solve for \( n \)
Subtract 24 from both sides:
\[ 7n = 52 - 24 \]
\[ 7n = 28 \]
Now, divide by 7:
\[ n = 4 \]

### Conclusion
Thus, the value of \( n \) is:
\[ \boxed{4} \]