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).

### Step 1: Write down the known values
We are given:
- First term \( a = 1 \)
- nth term \( a_n = 20 \)
- Sum of the first n terms \( S_n = 399 \)

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term of an AP is:
\[ a_n = a + (n - 1)d \]
Substituting the known values:
\[ 20 = 1 + (n - 1)d \]
This simplifies to:
\[ 19 = (n - 1)d \quad \text{(Equation 1)} \]

### Step 3: Use the formula for the sum of the first n terms
The formula for the sum of the first n terms of an AP is:
\[ S_n = \frac{n}{2} (a + a_n) \]
Substituting the known values:
\[ 399 = \frac{n}{2} (1 + 20) \]
This simplifies to:
\[ 399 = \frac{n}{2} \times 21 \]
Multiplying both sides by 2:
\[ 798 = 21n \]
Dividing both sides by 21:
\[ n = \frac{798}{21} \quad \text{(Equation 2)} \]

### Step 4: Calculate the value of n
Now, we will calculate \( n \):
\[ n = \frac{798}{21} \]
To simplify \( \frac{798}{21} \):
- First, divide 798 by 3 (the common factor of 21 and 798):
\[ 798 \div 3 = 266 \quad \text{and} \quad 21 \div 3 = 7 \]
So,
\[ n = \frac{266}{7} \]
Now we divide 266 by 7:
- 7 goes into 26 three times (3 × 7 = 21), leaving a remainder of 5.
- Bring down the 6 to make it 56. 7 goes into 56 eight times (8 × 7 = 56), leaving no remainder.

Thus,
\[ n = 38 \]

### Final Answer
The value of \( n \) is \( 38 \).

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