1. a) The first four terms of an infinite arithmetic sequence are:

$$
-\frac{3}{2}, \frac{1}{2}, \frac{5}{2}, \frac{9}{2}
$$

Calculate which number is the two hundredth term of this sequence and what is the sum of the first two hundred terms of the sequence.

b) Between the numbers $\frac{1}{2}$ and $\frac{9}{2}$, insert three numbers so that we get a geometric sequence. Determine all possible values of the inserted terms. The values should be exact. (6 points)
1. 

a) We calculate the difference of the arithmetic sequence \(d=\frac{1}{2}-\left(-\frac{3}{2}\right)=2\). We calculate the two hundredth term of the sequence \(a_{200}=a_{1}+(200-1) \cdot d=\frac{793}{2}\). We calculate the sum of the first two hundred terms \(s_{200}=\frac{200}{2}\left(a_{1}+a_{200}\right)=\frac{200}{2}\left(-\frac{3}{2}+\frac{793}{2}\right)=39500\).

b) Between the numbers \(\frac{1}{2}\) and \(\frac{9}{2}\), we insert three numbers \(a_{2}, a_{3}, a_{4}\). We get the sequence \(\frac{1}{2}, a_{2}, a_{3}, a_{4}, \frac{9}{2}\). The number \(\frac{9}{2}\) is the fifth term of the geometric sequence, so the equation \(a_{1} \cdot q^{4}=\frac{9}{2}\) holds. Substituting \(a_{1}\) into the equation, we get \(\frac{1}{2} \cdot q^{4}=\frac{9}{2}\). After eliminating the fractions, we get the equation \(q^{4}=9\). The equation has two solutions \(q_{1}=\sqrt{3}, q_{2}=-\sqrt{3}\). We write both solutions: \(\frac{\sqrt{3}}{2}, \frac{3}{2}, \frac{3 \sqrt{3}}{2} ; \quad-\frac{\sqrt{3}}{2}, \frac{3}{2},-\frac{3 \sqrt{3}}{2}\).

```
Calculation \(d=\frac{1}{2}-\left(-\frac{3}{2}\right)=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots\)

```

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=57&width=1728&top_left_y=914&top_left_x=178)

```
Calculation \(s_{200}=\frac{200}{2}\left(a_{1}+a_{200}\right)=\frac{200}{2}\left(-\frac{3}{2}+\frac{793}{2}\right)=39500 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots\)

```

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=60&width=1727&top_left_y=1009&top_left_x=176)

```

```

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=51&width=1727&top_left_y=1065&top_left_x=173)

```

```

![](https://cdn.mathpix.com/cropped/2024_06_07_18400104441c5ee1e47ag-19.jpg?height=49&width=1725&top_left_y=1112&top_left_x=177)

```
Writing the solutions \(\frac{\sqrt{3}}{2}, \frac{3}{2}, \frac{3 \sqrt{3}}{2} ; \quad-\frac{\sqrt{3}}{2}, \frac{3}{2},-\frac{3 \sqrt{3}}{2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots\) points
```