Find the first five terms of the sequence a sub n=((-1)^n+1)(n-1/2n+1).

Let's find the first five terms of the sequence given by the formula:

\[
a_n = \left((-1)^n + 1\right) \left(\frac{n - \frac{1}{2}}{n + 1}\right)
\]

We will calculate each term \( a_1, a_2, a_3, a_4, a_5 \) by substituting \( n = 1, 2, 3, 4, 5 \) into the formula:

1. **For \( n = 1 \):**

\[
a_1 = \left((-1)^1 + 1\right) \left(\frac{1 - \frac{1}{2}}{1 + 1}\right) = \left(-1 + 1\right) \left(\frac{\frac{1}{2}}{2}\right) = 0 \times \frac{1}{4} = 0
\]

2. **For \( n = 2 \):**

\[
a_2 = \left((-1)^2 + 1\right) \left(\frac{2 - \frac{1}{2}}{2 + 1}\right) = \left(1 + 1\right) \left(\frac{\frac{3}{2}}{3}\right) = 2 \times \frac{3}{6} = 2 \times \frac{1}{2} = 1
\]

3. **For \( n = 3 \):**

\[
a_3 = \left((-1)^3 + 1\right) \left(\frac{3 - \frac{1}{2}}{3 + 1}\right) = \left(-1 + 1\right) \left(\frac{\frac{5}{2}}{4}\right) = 0 \times \frac{5}{8} = 0
\]

4. **For \( n = 4 \):**

\[
a_4 = \left((-1)^4 + 1\right) \left(\frac{4 - \frac{1}{2}}{4 + 1}\right) = \left(1 + 1\right) \left(\frac{\frac{7}{2}}{5}\right) = 2 \times \frac{7}{10} = \frac{14}{10} = 1.4
\]

5. **For \( n = 5 \):**

\[
a_5 = \left((-1)^5 + 1\right) \left(\frac{5 - \frac{1}{2}}{5 + 1}\right) = \left(-1 + 1\right) \left(\frac{\frac{9}{2}}{6}\right) = 0 \times \frac{9}{12} = 0
\]

### First Five Terms:
The first five terms of the sequence are:

\[
a_1 = 0, \quad a_2 = 1, \quad a_3 = 0, \quad a_4 = 1.4, \quad a_5 = 0
\]

Would you like further details or have any questions?

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**Related Questions:**

1. What pattern do you notice in the sequence's terms as \( n \) increases?
2. How would the sequence change if \( (-1)^n + 1 \) were replaced by \( (-1)^{n+1} + 1 \)?
3. Can this sequence be related to any known mathematical series?
4. What is the behavior of the sequence as \( n \) approaches infinity?
5. How would the sequence change if the denominator of the fraction inside the sequence formula were \( n-1 \) instead of \( n+1 \)?

**Tip:** Notice how sequences with alternating terms (involving \( (-1)^n \)) often produce terms that either cancel out or follow a repeating pattern. This can be useful in predicting future terms without direct computation.

Discover the first five terms of the sequence given by the formula (-1)^n+1(n-1/2n+1), including detailed calculations for each term. Understand the alternating pattern and its mathematical implications. Suitable for advanced high school students and beyond.

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