15th term of the A.P.`1/3, 5/3, 9/3, 13/3………………,` is :

Options

A. `61/3`
B. `6`
C. `5`
D. `19`

Answer

To find the 15th term of the arithmetic progression (A.P.) given by the sequence \( \frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \ldots \), we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) of the A.P. is:
\[ a = \frac{1}{3} \]

To find the common difference \( d \), we subtract the first term from the second term:
\[ d = \text{second term} - \text{first term} = \frac{5}{3} - \frac{1}{3} \]
Calculating this gives:
\[ d = \frac{5 - 1}{3} = \frac{4}{3} \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term \( T_n \) of an A.P. is given by:
\[ T_n = a + (n - 1) \cdot d \]
Here, we need to find the 15th term, so \( n = 15 \).

### Step 3: Substitute the values into the formula
Substituting \( a \), \( d \), and \( n \) into the formula:
\[ T_{15} = \frac{1}{3} + (15 - 1) \cdot \frac{4}{3} \]
This simplifies to:
\[ T_{15} = \frac{1}{3} + 14 \cdot \frac{4}{3} \]

### Step 4: Calculate \( 14 \cdot \frac{4}{3} \)
Calculating \( 14 \cdot \frac{4}{3} \):
\[ 14 \cdot \frac{4}{3} = \frac{56}{3} \]

### Step 5: Add the two fractions
Now, we add \( \frac{1}{3} \) and \( \frac{56}{3} \):
\[ T_{15} = \frac{1}{3} + \frac{56}{3} = \frac{1 + 56}{3} = \frac{57}{3} \]

### Step 6: Simplify the fraction
Now we simplify \( \frac{57}{3} \):
\[ \frac{57}{3} = 19 \]

### Final Answer
Thus, the 15th term of the A.P. is:
\[ \boxed{19} \]