In an AP, if `a = 1`, `a_(n)= 20` and `S_(n) = 399`, then `n` is equal to

Options

A. 19
B. 21
C. 38
D. 42

Explanation

To solve the problem step by step, we will use the formulas for the nth term and the sum of the first n terms of an arithmetic progression (AP).

### Given:
- First term \( a = 1 \)
- \( a_n = 20 \) (nth term)
- \( S_n = 399 \) (sum of the first n terms)

### Step 1: Use the formula for the nth term of an AP
The formula for the nth term \( a_n \) of an AP is given by:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 20 = 1 + (n - 1) \cdot d \]
This simplifies to:
\[ 20 - 1 = (n - 1) \cdot d \]
\[ 19 = (n - 1) \cdot d \quad \text{(Equation 1)} \]

### Step 2: Use the formula for the sum of the first n terms of an AP
The formula for the sum \( S_n \) of the first n terms is:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]
Substituting the known values:
\[ 399 = \frac{n}{2} \cdot (2 \cdot 1 + (n - 1) \cdot d) \]
This simplifies to:
\[ 399 = \frac{n}{2} \cdot (2 + (n - 1) \cdot d) \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 798 = n \cdot (2 + (n - 1) \cdot d) \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 1 into Equation 2
From Equation 1, we have:
\[ d = \frac{19}{n - 1} \]
Now substitute \( d \) into Equation 2:
\[ 798 = n \cdot \left(2 + (n - 1) \cdot \frac{19}{n - 1}\right) \]
This simplifies to:
\[ 798 = n \cdot (2 + 19) \]
\[ 798 = n \cdot 21 \]

### Step 4: Solve for n
Now, divide both sides by 21:
\[ n = \frac{798}{21} \]
Calculating this gives:
\[ n = 38 \]

### Final Answer
Thus, the value of \( n \) is \( \boxed{38} \).