The sum of first n terms of an AP is `(5n - n^(2)).` The nth term of the AP is

Options

A. (5-2n)
B. (6-2n)
C. (2n -5)
D. (2n-6)

Explanation

To find the nth term of the arithmetic progression (AP) given the sum of the first n terms \( S_n = 5n - n^2 \), we can follow these steps:

### Step 1: Calculate \( S_1 \) and \( S_2 \)
To find the first term \( a_1 \) of the AP, we need to calculate \( S_1 \) and \( S_2 \).

- For \( n = 1 \):
\[ S_1 = 5(1) - (1)^2 = 5 - 1 = 4 \]

- For \( n = 2 \):
\[ S_2 = 5(2) - (2)^2 = 10 - 4 = 6 \]

### Step 2: Find the first term \( a_1 \)
The first term \( a_1 \) is simply \( S_1 \):
\[ a_1 = S_1 = 4 \]

### Step 3: Calculate \( S_3 \)
Next, we calculate \( S_3 \) to find the second term \( a_2 \).

- For \( n = 3 \):
\[ S_3 = 5(3) - (3)^2 = 15 - 9 = 6 \]

### Step 4: Find the second term \( a_2 \)
The second term \( a_2 \) can be found using:
\[ a_2 = S_2 - S_1 = 6 - 4 = 2 \]

### Step 5: Calculate \( S_4 \)
Now we calculate \( S_4 \) to find the third term \( a_3 \).

- For \( n = 4 \):
\[ S_4 = 5(4) - (4)^2 = 20 - 16 = 4 \]

### Step 6: Find the third term \( a_3 \)
The third term \( a_3 \) can be found using:
\[ a_3 = S_3 - S_2 = 6 - 6 = 0 \]

### Step 7: Identify the common difference
Now we have the first three terms of the AP:
- \( a_1 = 4 \)
- \( a_2 = 2 \)
- \( a_3 = 0 \)

To find the common difference \( d \):
\[ d = a_2 - a_1 = 2 - 4 = -2 \]

### Step 8: Write the formula for the nth term
The nth term \( a_n \) of an AP is given by:
\[ a_n = a_1 + (n - 1)d \]
Substituting the values we have:
\[ a_n = 4 + (n - 1)(-2) \]

### Step 9: Simplify the expression
Now simplify the expression:
\[ a_n = 4 - 2(n - 1) = 4 - 2n + 2 = 6 - 2n \]

### Final Answer
Thus, the nth term of the AP is:
\[ \boxed{6 - 2n} \]