Find the sum of the first 20 terms 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 by the sequence 5, 8, 11, 14, ..., we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
- The first term \( A \) is the first number in the sequence, which is 5.
- The common difference \( D \) can be found 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:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]
Where:
- \( n \) is the number of terms (in this case, 20),
- \( A \) is the first term,
- \( D \) is the common difference.

### 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
- First, calculate \( \frac{20}{2} \):
\[ \frac{20}{2} = 10 \]
- Next, calculate \( 2 \times 5 \):
\[ 2 \times 5 = 10 \]
- Then calculate \( (20 - 1) \times 3 \):
\[ (20 - 1) \times 3 = 19 \times 3 = 57 \]
- Now, combine these results:
\[ S_{20} = 10 \times (10 + 57) = 10 \times 67 \]

### Step 5: Final calculation
- Finally, calculate \( 10 \times 67 \):
\[ S_{20} = 670 \]

### Conclusion
The sum of the first 20 terms of the A.P. is \( 670 \).

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