If first and eightth terms of a G.P. are ` x^(-4) and x^(52)`
and its second term is ` x^(t)` then t is equal to

Options

A. 2
B. 3
C. 4
D. 13

Answer

To solve the problem step by step, we will use the properties of a geometric progression (G.P.).

### Step-by-Step Solution:

1. **Identify the terms of the G.P.**:
- The first term \( a \) is given as \( x^{-4} \).
- The eighth term \( a_8 \) is given as \( x^{52} \).

2. **Use the formula for the \( n \)-th term of a G.P.**:
- The \( n \)-th term of a G.P. can be expressed as:
\[ a_n = a \cdot r^{n-1} \]
- For the eighth term:
\[ a_8 = a \cdot r^{7} \]
- Substituting the values:
\[ x^{52} = x^{-4} \cdot r^{7} \]

3. **Rearranging the equation**:
- We can rearrange the equation to isolate \( r^{7} \):
\[ r^{7} = x^{52} \cdot x^{4} = x^{52 + 4} = x^{56} \]

4. **Finding the common ratio \( r \)**:
- To find \( r \), take the seventh root of both sides:
\[ r = (x^{56})^{\frac{1}{7}} = x^{\frac{56}{7}} = x^{8} \]

5. **Finding the second term**:
- The second term \( a_2 \) is given by:
\[ a_2 = a \cdot r^{1} = x^{-4} \cdot x^{8} \]
- Simplifying this:
\[ a_2 = x^{-4 + 8} = x^{4} \]

6. **Setting the second term equal to \( x^{t} \)**:
- We know from the problem that the second term is also given as \( x^{t} \):
\[ x^{t} = x^{4} \]

7. **Comparing the exponents**:
- Since the bases are the same, we can equate the exponents:
\[ t = 4 \]

### Final Answer:
Thus, the value of \( t \) is \( 4 \).

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