Let's solve the given problems regarding geometric series. For each part, we'll calculate the specified sum $S_n$, where $n$ is the number of terms.

a) $S_{10}$ for $\sqrt{3} - 3 + 3\sqrt{3} + \ldots$

This is a geometric series with:

• First term $a = \sqrt{3}$,
• Common ratio $r = -\frac{3}{\sqrt{3}} = -\sqrt{3}$.

The sum of the first $n$ terms of a geometric series is: $S_n = a \frac{1 - r^n}{1 - r}, \quad \text{if } r \neq 1.$

Substituting $a = \sqrt{3}$, $r = -\sqrt{3}$, and $n = 10$: $S_{10} = \sqrt{3} \frac{1 - (-\sqrt{3})^{10}}{1 - (-\sqrt{3})}.$

Simplify the denominator: $1 - (-\sqrt{3}) = 1 + \sqrt{3}.$

Now compute $(-\sqrt{3})^{10}$: $(-\sqrt{3})^{10} = (\sqrt{3})^{10} \cdot (-1)^{10} = (\sqrt{3})^{10}.$

So, $S_{10} = \sqrt{3} \frac{1 - (\sqrt{3})^{10}}{1 + \sqrt{3}}.$

b) $S_{12}$ for $\sqrt{2}x + 2x + 2\sqrt{2}x + \ldots$

This is a geometric series with:

• First term $a = \sqrt{2}x$,
• Common ratio $r = \frac{2x}{\sqrt{2}x} = \sqrt{2}$.

Using the sum formula $S_n = a \frac{1 - r^n}{1 - r}$, we substitute $a = \sqrt{2}x$, $r = \sqrt{2}$, and $n = 12$: $S_{12} = \sqrt{2}x \frac{1 - (\sqrt{2})^{12}}{1 - \sqrt{2}}.$

Simplify the denominator: $1 - \sqrt{2}.$

So, $S_{12} = \sqrt{2}x \frac{1 - (\sqrt{2})^{12}}{1 - \sqrt{2}}.$

c) $S_n$ for $2 + 2x + 2x^2 + \ldots + 2x^{n-1}$

This is a geometric series with:

• First term $a = 2$,
• Common ratio $r = x$.

The sum formula $S_n = a \frac{1 - r^n}{1 - r}$ applies: $S_n = 2 \frac{1 - x^n}{1 - x}, \quad \text{if } x \neq 1.$

Would you like detailed numerical examples for $S_{10}$ and $S_{12}$, or further clarification? 😊

Relative Questions

1. How do we compute powers of roots like $(\sqrt{3})^{10}$?
2. How does the sum formula change if the common ratio $r$ is 1?
3. What happens if $|r| > 1$ in a geometric series?
4. How can geometric series be applied to model financial growth or decay?
5. Can the sum formula be used for infinite series? If so, under what conditions?

Tip

When solving geometric series, always double-check the common ratio by dividing the second term by the first term!