In an A.P. find `S_(n)` where `a_(n) = 5n-1`. Hence find the sum of the first 20 terms.

Answer

To solve the problem, we need to find the sum of the first `n` terms of the arithmetic progression (A.P.) defined by the formula \( a_n = 5n - 1 \).

### Step 1: Identify the first term \( a_1 \)
The first term \( a_1 \) can be found by substituting \( n = 1 \) into the formula:
\[ a_1 = a(1) = 5(1) - 1 = 5 - 1 = 4 \]

### Step 2: Identify the last term \( a_n \)
The last term \( a_n \) is given by the formula:
\[ a_n = a(n) = 5n - 1 \]

### Step 3: Write the formula for the sum of the first \( n \) terms \( S_n \)
The formula for the sum of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
Substituting \( a_1 \) and \( a_n \):
\[ S_n = \frac{n}{2} \times (4 + (5n - 1)) \]
This simplifies to:
\[ S_n = \frac{n}{2} \times (4 + 5n - 1) = \frac{n}{2} \times (5n + 3) \]

### Step 4: Simplify \( S_n \)
Now we can simplify \( S_n \):
\[ S_n = \frac{n(5n + 3)}{2} \]

### Step 5: Calculate the sum of the first 20 terms \( S_{20} \)
To find the sum of the first 20 terms, substitute \( n = 20 \) into the formula:
\[ S_{20} = \frac{20(5 \times 20 + 3)}{2} \]
Calculating inside the parentheses:
\[ 5 \times 20 = 100 \quad \text{so} \quad 5 \times 20 + 3 = 100 + 3 = 103 \]
Now substituting back:
\[ S_{20} = \frac{20 \times 103}{2} = \frac{2060}{2} = 1030 \]

### Final Answer
The sum of the first 20 terms of the A.P. is \( S_{20} = 1030 \).

---