15th term of 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 terms \( \frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3} \), we can follow these steps:

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

Next, we calculate the common difference \( d \) by subtracting the first term from the second term:
\[ d = \frac{5}{3} - \frac{1}{3} = \frac{4}{3} \]

### 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:
\[ T_n = a + (n - 1) \cdot d \]
where \( n \) is the term number.

### Step 3: Substitute n = 15 into the formula
We want to find the 15th term, so we substitute \( n = 15 \):
\[ T_{15} = a + (15 - 1) \cdot d \]
\[ T_{15} = \frac{1}{3} + (14) \cdot \frac{4}{3} \]

### Step 4: Simplify the expression
Calculating \( (14) \cdot \frac{4}{3} \):
\[ T_{15} = \frac{1}{3} + \frac{56}{3} \]
Now, combine the two fractions:
\[ T_{15} = \frac{1 + 56}{3} = \frac{57}{3} \]

### Step 5: Final result
Now, simplify \( \frac{57}{3} \):
\[ T_{15} = 19 \]

Thus, the 15th term of the A.P. is \( 19 \).

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