Find the sum of the first 20 term of the A.P. 5,8,11,14……….

Answer

To find the sum of the first 20 terms of the arithmetic progression (A.P.) given as 5, 8, 11, 14, ..., we will follow these steps:

### Step 1: Identify the first term and the common difference
The first term \( a \) of the A.P. is the first number in the sequence:
\[ a = 5 \]

Next, we find the common difference \( d \) by subtracting the first term from the second term:
\[ d = a_2 - a_1 = 8 - 5 = 3 \]

### 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:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

In this case, we need to find \( S_{20} \) (the sum of the first 20 terms), so we set \( n = 20 \).

### Step 3: Substitute the values into the formula
Substituting \( n = 20 \), \( a = 5 \), and \( d = 3 \) into the formula:
\[ S_{20} = \frac{20}{2} \times (2 \times 5 + (20 - 1) \times 3) \]

### Step 4: Simplify the expression
Calculating the components:
1. \( \frac{20}{2} = 10 \)
2. \( 2 \times 5 = 10 \)
3. \( 20 - 1 = 19 \)
4. \( 19 \times 3 = 57 \)

Now substitute these values back into the formula:
\[ S_{20} = 10 \times (10 + 57) \]

### Step 5: Final calculation
Now, calculate \( 10 + 57 \):
\[ 10 + 57 = 67 \]
Then multiply by 10:
\[ S_{20} = 10 \times 67 = 670 \]

### Conclusion
The sum of the first 20 terms of the A.P. is:
\[ \boxed{670} \]