Find the sum of n terms of an A.P. whose nth term is (2n+1).

Answer

To find the sum of the first n terms of an 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 (a) and the common difference (d)
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 find 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:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step 3: Substitute the values of a and d into the formula
Now, substituting \( a = 3 \) and \( d = 2 \) into the sum formula:
\[ S_n = \frac{n}{2} \times (2(3) + (n - 1)(2)) \]

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

### Step 5: Factor out the common terms
Factoring out the 2:
\[ S_n = \frac{n}{2} \times 2(n + 2) \]
\[ = n(n + 2) \]

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

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