Let's solve both geometric series problems from the image.

Formula for the sum of a finite geometric series:

$S_n = \frac{a_1 (1 - r^n)}{1 - r}$ where:

• $S_n$ is the sum of the first $n$ terms,
• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the number of terms.

Problem 1:

Find the sum of the first six terms of the geometric sequence: 3, 6, 12, 24, ...

Step 1: Identify the variables.

• $a_1 = 3$ (the first term),
• $r = \frac{6}{3} = 2$ (the common ratio),
• $n = 6$ (the number of terms).

Step 2: Use the formula. $S_6 = \frac{3(1 - 2^6)}{1 - 2}$ $S_6 = \frac{3(1 - 64)}{-1}$ $S_6 = \frac{3(-63)}{-1} = 189$ So, the sum of the first six terms is 189.

Problem 2:

Find the sum of 10 terms of the sequence: 1, $\frac{1}{2}$, $\frac{1}{4}$, ...

Step 1: Identify the variables.

• $a_1 = 1$ (the first term),
• $r = \frac{\frac{1}{2}}{1} = \frac{1}{2}$ (the common ratio),
• $n = 10$ (the number of terms).

Step 2: Use the formula. $S_{10} = \frac{1(1 - \left(\frac{1}{2}\right)^{10})}{1 - \frac{1}{2}}$ $S_{10} = \frac{1(1 - \frac{1}{1024})}{\frac{1}{2}}$ $S_{10} = \frac{1(1 - 0.0009765625)}{0.5} = \frac{0.9990234375}{0.5}$ $S_{10} = 1.998046875$ So, the sum of the first 10 terms is approximately 1.998.

Do you need any further clarifications or more details on these steps?

Here are 5 questions to expand on this:

1. What happens if the common ratio $r$ is greater than 1 for an infinite geometric series?
2. How does the sum of a geometric series change if the common ratio is negative?
3. How can you calculate the sum of an infinite geometric series when $|r| < 1$?
4. How is the geometric series applied in real-world financial models?
5. Can you think of an example where a geometric sequence models exponential growth?

Tip: For an infinite geometric series where $|r| < 1$, the sum formula simplifies to $\frac{a_1}{1 - r}$.