Find the sum of:
n terms of 4, 7 , 10, ....

Answer

To find the sum of the first \( n \) terms of the arithmetic progression (AP) series given by 4, 7, 10, ..., we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( A \) of the series is:
\[ A = 4 \]
The common difference \( D \) can be calculated as:
\[ D = 7 - 4 = 3 \]
or
\[ D = 10 - 7 = 3 \]
Thus, the common difference \( D \) is:
\[ D = 3 \]

### Step 2: Use the formula for the sum of the first \( n \) terms of an AP
The formula for the sum of the first \( n \) terms \( S_n \) of an arithmetic progression 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
Substituting \( A = 4 \) and \( D = 3 \) into the formula, we get:
\[ S_n = \frac{n}{2} \times (2 \times 4 + (n - 1) \times 3) \]

### Step 4: Simplify the expression
Now, simplify the expression inside the parentheses:
\[ S_n = \frac{n}{2} \times (8 + (n - 1) \times 3) \]
Distributing \( 3 \) in the second term:
\[ S_n = \frac{n}{2} \times (8 + 3n - 3) \]
Combine like terms:
\[ S_n = \frac{n}{2} \times (3n + 5) \]

### Step 5: Final expression for the sum of the first \( n \) terms
Thus, the sum of the first \( n \) terms of the series is:
\[ S_n = \frac{n}{2} \times (3n + 5) \]

### Summary
The sum of the first \( n \) terms of the series 4, 7, 10, ... is:
\[ S_n = \frac{n}{2} (3n + 5) \]