Find the sum of first n term of the A.P., whose nth term is given by (2n+1).

Answer

To find the sum of the first n terms of the arithmetic progression (A.P.) whose nth term is given by \( a_n = 2n + 1 \), we can follow these steps:

### Step 1: Identify the first term and the common difference
The nth term of the A.P. is given by:
\[ a_n = 2n + 1 \]
To find the first term \( a_1 \):
\[ a_1 = a(1) = 2(1) + 1 = 3 \]

To find the second term \( a_2 \):
\[ a_2 = a(2) = 2(2) + 1 = 5 \]

Now, we can calculate the common difference \( d \):
\[ d = a_2 - a_1 = 5 - 3 = 2 \]

### Step 2: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first n terms \( S_n \) of an A.P. is given by:
\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \]
Substituting the values of \( a \) and \( d \):
\[ S_n = \frac{n}{2} \left(2(3) + (n - 1)(2)\right) \]

### Step 3: Simplify the expression
Calculating inside the parentheses:
\[ S_n = \frac{n}{2} \left(6 + 2(n - 1)\right) \]
\[ = \frac{n}{2} \left(6 + 2n - 2\right) \]
\[ = \frac{n}{2} \left(2n + 4\right) \]

### Step 4: Factor out the common terms
\[ S_n = \frac{n}{2} \cdot 2(n + 2) \]
\[ = n(n + 2) \]

### Final Result
Thus, the sum of the first n terms of the A.P. is:
\[ S_n = n^2 + 2n \]

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