The sum of the first n terms of an arithmetic series whose nth term is n can be calculate by the formula `n(n + )//2`. Which one of the following equals the first eight terms in a series whose nth terms is 2n?

Options

A. 24
B. 48
C. 56
D. 72

Answer

To solve the problem, we need to find the sum of the first eight terms of a series whose nth term is given by \(2n\).

### Step-by-Step Solution:

1. **Identify the nth term**:
The nth term of the series is given as \(a_n = 2n\).

2. **Calculate the first eight terms**:
We can find the first eight terms by substituting \(n\) from 1 to 8 into the formula \(a_n = 2n\):
- For \(n = 1\): \(a_1 = 2 \times 1 = 2\)
- For \(n = 2\): \(a_2 = 2 \times 2 = 4\)
- For \(n = 3\): \(a_3 = 2 \times 3 = 6\)
- For \(n = 4\): \(a_4 = 2 \times 4 = 8\)
- For \(n = 5\): \(a_5 = 2 \times 5 = 10\)
- For \(n = 6\): \(a_6 = 2 \times 6 = 12\)
- For \(n = 7\): \(a_7 = 2 \times 7 = 14\)
- For \(n = 8\): \(a_8 = 2 \times 8 = 16\)

Therefore, the first eight terms are: \(2, 4, 6, 8, 10, 12, 14, 16\).

3. **Sum the first eight terms**:
Now, we need to find the sum of these terms:
\[ S_8 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 \]
\[ S_8 = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 \]

We can also use the formula for the sum of the first \(n\) terms of an arithmetic series:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
Here, \(n = 8\), \(a_1 = 2\), and \(a_8 = 16\):
\[ S_8 = \frac{8}{2} \times (2 + 16) = 4 \times 18 = 72 \]

4. **Conclusion**:
The sum of the first eight terms of the series whose nth term is \(2n\) is \(72\).

### Final Answer:
The answer is \(72\).