Good morning sa lahat. Kindly answer the question below.(Notebook).Sundan nio lang yung naisend kong topic kahapon.
    Find the sum to infinity of each geometric sequence.
1) 64,16, 4,1...

2) 1/3, 1/9, 1/27, 1/81...

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 \( \frac{1}{2} \).

---

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!

Good morning sa lahat. Kindly answer the question below.(Notebook).Sundan nio lang yung naisend kong topic kahapon. Find the sum to infinity of each geometric sequence. 1) 64,16, 4,1... 2) 1/3, 1/9, 1/27, 1/81...

Learn how to find the sum to infinity for geometric sequences such as 64, 16, 4, 1... and 1/3, 1/9, 1/27... Using the sum to infinity formula, we calculate results for these sequences. This detailed explanation is suitable for students in Grades 9-11.

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