Write the first five terms of the sequences in whose nth terms is :
`a_(n)=(-1)^(n-1).5^(n+1)`

Options

A. 5 terms of given sequence are
`a_(1)=25,a_(2)=-125,a_(3)=625,a_(4)=-3125,a_(5)=15625.`
B. 5 terms of given sequence are
`a_(1)=25,a_(2)=125,a_(3)=625,a_(4)=3125,a_(5)=15625.`
C. 5 terms of given sequence are
`a_(1)=-25,a_(2)=125,a_(3)=--625,a_(4)=3125,a_(5)=-15625.`
D. None of these

Explanation

To find the first five terms of the sequence defined by the nth term \( a_n = (-1)^{n-1} \cdot 5^{n+1} \), we will calculate \( a_1, a_2, a_3, a_4, \) and \( a_5 \) step by step.

### Step 1: Calculate \( a_1 \)
Using the formula:
\[ a_1 = (-1)^{1-1} \cdot 5^{1+1} \]
Calculating:
\[ a_1 = (-1)^{0} \cdot 5^{2} = 1 \cdot 25 = 25 \]

### Step 2: Calculate \( a_2 \)
Using the formula:
\[ a_2 = (-1)^{2-1} \cdot 5^{2+1} \]
Calculating:
\[ a_2 = (-1)^{1} \cdot 5^{3} = -1 \cdot 125 = -125 \]

### Step 3: Calculate \( a_3 \)
Using the formula:
\[ a_3 = (-1)^{3-1} \cdot 5^{3+1} \]
Calculating:
\[ a_3 = (-1)^{2} \cdot 5^{4} = 1 \cdot 625 = 625 \]

### Step 4: Calculate \( a_4 \)
Using the formula:
\[ a_4 = (-1)^{4-1} \cdot 5^{4+1} \]
Calculating:
\[ a_4 = (-1)^{3} \cdot 5^{5} = -1 \cdot 3125 = -3125 \]

### Step 5: Calculate \( a_5 \)
Using the formula:
\[ a_5 = (-1)^{5-1} \cdot 5^{5+1} \]
Calculating:
\[ a_5 = (-1)^{4} \cdot 5^{6} = 1 \cdot 15625 = 15625 \]

### Final Result
The first five terms of the sequence are:
\[ a_1 = 25, \quad a_2 = -125, \quad a_3 = 625, \quad a_4 = -3125, \quad a_5 = 15625 \]