Let's go through the questions in the image step by step.

1. Find the specified sum for each of the following geometric series:

a. $5 + 10 + 20 + \dots$ find $S_{10}$

This is a geometric series with:

• First term $a = 5$
• Common ratio $r = \frac{10}{5} = 2$

The sum of the first $n$ terms of a geometric series is given by the formula: $S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{(for } r \neq 1)$ For $S_{10}$, we plug in:

• $a = 5$
• $r = 2$
• $n = 10$ $S_{10} = \frac{5(1 - 2^{10})}{1 - 2}$

b. $1 - 3 + 9 - \dots$ find $S_6$

This is another geometric series with alternating signs:

• First term $a = 1$
• Common ratio $r = -3$

Again, using the formula for the sum of the first $n$ terms: $S_6 = \frac{1(1 - (-3)^6)}{1 - (-3)}$

c. $\frac{4}{3} + \frac{2}{3} + \frac{1}{3} + \dots$ find $S_5$

This geometric series has:

• First term $a = \frac{4}{3}$
• Common ratio $r = \frac{1}{2}$

Applying the sum formula: $S_5 = \frac{\frac{4}{3}(1 - \left(\frac{1}{2}\right)^5)}{1 - \frac{1}{2}}$

2. For the geometric sequence $2, 6, 18, \dots$ write an expression for:

a. The $n$-th term

In a geometric sequence, the $n$-th term is given by: $a_n = a \cdot r^{n-1}$ Here, $a = 2$ and $r = \frac{6}{2} = 3$. So the expression for the $n$-th term is: $a_n = 2 \cdot 3^{n-1}$

b. The sum of the first $n$ terms

Using the sum formula again: $S_n = \frac{a(1 - r^n)}{1 - r}$ For this series, $a = 2$ and $r = 3$, so: $S_n = \frac{2(1 - 3^n)}{1 - 3}$

3. For the geometric sequence $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$, write an expression for:

a. The $n$-th term

Here:

• First term $a = 1$
• Common ratio $r = \frac{1}{2}$

So the $n$-th term is: $a_n = 1 \cdot \left(\frac{1}{2}\right)^{n-1}$ Simplifying: $a_n = \left(\frac{1}{2}\right)^{n-1}$

b. The sum of the first $n$ terms

Using the sum formula for geometric series: $S_n = \frac{1(1 - \left(\frac{1}{2}\right)^n)}{1 - \frac{1}{2}}$ Simplifying: $S_n = 2 \left(1 - \left(\frac{1}{2}\right)^n\right)$

Would you like a detailed breakdown of any particular solution or calculation?

Here are 5 relative questions for further practice:

1. How would the sum formula change if the common ratio is negative?
2. Can you find $S_5$ for the sequence $1 + 2 + 4 + 8 + 16 + \dots$?
3. For the series $2, 6, 18, 54, \dots$, calculate the sum of the first 7 terms.
4. How would you calculate the 12th term of the series $3 + 6 + 12 + 24 + \dots$?
5. What happens to the sum of an infinite geometric series if $|r| < 1$?

Tip: When working with geometric series, always check if the common ratio is between -1 and 1, as this allows for the sum of an infinite series to exist.