Good morning! Let's solve each of the geometric sequence problems one by one.

Formula for the Sum to Infinity of a Geometric Sequence:

$S_{\infty} = \frac{a}{1 - r}$ where:

• $S_{\infty}$ is the sum to infinity,
• $a$ is the first term of the sequence,
• $r$ is the common ratio, and $|r| < 1$.

1) Sequence: $64, 16, 4, 1, \dots$

• First term ($a$) = 64
• Common ratio ($r$) = $\frac{16}{64} = \frac{1}{4}$

Now, applying the sum to infinity formula:

$S_{\infty} = \frac{64}{1 - \frac{1}{4}} = \frac{64}{\frac{3}{4}} = 64 \times \frac{4}{3} = \frac{256}{3}$

Thus, the sum to infinity is $\frac{256}{3}$ or approximately 85.33.

2) Sequence: $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$

• First term ($a$) = $\frac{1}{3}$
• Common ratio ($r$) = $\frac{\frac{1}{9}}{\frac{1}{3}} = \frac{1}{3}$

Applying the sum to infinity formula:

$S_{\infty} = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}$

Thus, the sum to infinity is $\frac{1}{2}$.

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Would you like further details or have any questions?

Here are some related questions to explore:

1. What if the common ratio $r$ is greater than 1?
2. How do you find the sum of a finite geometric sequence?
3. Can the sum to infinity exist if $|r| \geq 1$?
4. What is the sum to infinity when the first term is negative?
5. How do geometric series relate to exponential decay?

Tip: Always check if the absolute value of the common ratio $|r|$ is less than 1, otherwise the sum to infinity does not exist!