If the sum of 'n' terms of an arithmetic progression is `n^(2)-2n`, then what is the `n^(th)` term?

Options

A. `3n-n^(2)`
B. `n2n-3`
C. `2n+3`
D. `2n-5`

Explanation

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

### Step 1: Write down the formula for the sum of n terms
The sum of the first n terms of an arithmetic progression is given by:
\[ S_n = n^2 - 2n \]

### Step 2: Find the sum of the first (n-1) terms
To find the nth term, we first need to calculate the sum of the first (n-1) terms, \( S_{n-1} \):
\[ S_{n-1} = (n-1)^2 - 2(n-1) \]
Expanding this:
\[ S_{n-1} = (n^2 - 2n + 1) - (2n - 2) = n^2 - 2n + 1 - 2n + 2 = n^2 - 4n + 3 \]

### Step 3: Use the formula for the nth term
The nth term \( T_n \) can be found using the formula:
\[ T_n = S_n - S_{n-1} \]
Substituting the values we have:
\[ T_n = (n^2 - 2n) - (n^2 - 4n + 3) \]
Simplifying this:
\[ T_n = n^2 - 2n - n^2 + 4n - 3 = 2n - 3 \]

### Conclusion
Thus, the nth term of the arithmetic progression is:
\[ T_n = 2n - 3 \]

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