The sum of n terms of an A.P. series is `(n^(2) + 2n)` for all values of n. Find the first 3 terms of the series:

Answer

To find the first three terms of the arithmetic progression (A.P.) series given that the sum of the first n terms \( S_n \) is \( n^2 + 2n \), we can follow these steps:

### Step 1: Find the first term \( a_1 \)
The sum of the first term \( S_1 \) can be calculated by substituting \( n = 1 \) into the given formula for \( S_n \):
\[ S_1 = 1^2 + 2 \cdot 1 = 1 + 2 = 3 \]
Thus, the first term \( a_1 \) is:
\[ a_1 = S_1 = 3 \]

### Step 2: Find the second term \( a_2 \)
Next, we calculate the sum of the first two terms \( S_2 \) by substituting \( n = 2 \):
\[ S_2 = 2^2 + 2 \cdot 2 = 4 + 4 = 8 \]
The sum of the first two terms can be expressed as:
\[ S_2 = a_1 + a_2 \]
Substituting \( a_1 = 3 \):
\[ 8 = 3 + a_2 \]
Solving for \( a_2 \):
\[ a_2 = 8 - 3 = 5 \]

### Step 3: Find the third term \( a_3 \)
Now, we find the sum of the first three terms \( S_3 \) by substituting \( n = 3 \):
\[ S_3 = 3^2 + 2 \cdot 3 = 9 + 6 = 15 \]
The sum of the first three terms can be expressed as:
\[ S_3 = a_1 + a_2 + a_3 \]
Substituting \( a_1 = 3 \) and \( a_2 = 5 \):
\[ 15 = 3 + 5 + a_3 \]
Solving for \( a_3 \):
\[ a_3 = 15 - (3 + 5) = 15 - 8 = 7 \]

### Conclusion
The first three terms of the A.P. series are:
\[ a_1 = 3, \quad a_2 = 5, \quad a_3 = 7 \]

### Summary
Thus, the first three terms of the series are \( 3, 5, 7 \).

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