If the first term in an arithmetic series is `8/3` and the last term is `(40)/(3)`, and the sum is 72, what are the first four terms ?

Options

A. `8/3,(12)/3,(6)/3,(20)/3`
B. `8/3,(14)/3,(20)/3,(26)/3`
C. `8/3,(16)/3,(24)/3,(32)/3`
D. `8/3,(16)/3,(32)/3,(40)/3`

Answer

To solve the problem, we need to find the first four terms of the arithmetic series given the first term, last term, and the sum of the series.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- First term \( a = \frac{8}{3} \)
- Last term \( l = \frac{40}{3} \)
- Sum of the series \( S_n = 72 \)

2. **Use the Sum Formula for Arithmetic Series:**
The formula for the sum of an arithmetic series is given by:
\[ S_n = \frac{n}{2} (a + l) \]
where \( n \) is the number of terms.

3. **Substitute the Known Values into the Formula:**
\[ 72 = \frac{n}{2} \left( \frac{8}{3} + \frac{40}{3} \right) \]
Simplifying the terms inside the parentheses:
\[ \frac{8}{3} + \frac{40}{3} = \frac{48}{3} = 16 \]
Now substituting back into the equation:
\[ 72 = \frac{n}{2} \cdot 16 \]

4. **Solve for \( n \):**
Multiply both sides by 2:
\[ 144 = 16n \]
Now divide both sides by 16:
\[ n = \frac{144}{16} = 9 \]
So, there are 9 terms in the series.

5. **Find the Common Difference \( d \):**
The last term of the arithmetic series can also be expressed as:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ \frac{40}{3} = \frac{8}{3} + (9-1)d \]
Simplifying:
\[ \frac{40}{3} = \frac{8}{3} + 8d \]
Rearranging gives:
\[ 8d = \frac{40}{3} - \frac{8}{3} = \frac{32}{3} \]
Now divide by 8:
\[ d = \frac{32}{3} \cdot \frac{1}{8} = \frac{4}{3} \]

6. **Calculate the First Four Terms:**
The first four terms of the arithmetic series are:
- First term: \( a = \frac{8}{3} \)
- Second term: \( a + d = \frac{8}{3} + \frac{4}{3} = \frac{12}{3} = 4 \)
- Third term: \( a + 2d = \frac{8}{3} + 2 \cdot \frac{4}{3} = \frac{8}{3} + \frac{8}{3} = \frac{16}{3} \)
- Fourth term: \( a + 3d = \frac{8}{3} + 3 \cdot \frac{4}{3} = \frac{8}{3} + \frac{12}{3} = \frac{20}{3} \)

### Final Answer:
The first four terms of the arithmetic series are:
1. \( \frac{8}{3} \)
2. \( 4 \)
3. \( \frac{16}{3} \)
4. \( \frac{20}{3} \)