write\:an\:explicit\:formula\:for\:an\:geometric\:sequence\:that\:has\:the\:points\:\left(1,\:8\right)\:and\:\left(2,\:4\right)

To find the explicit formula for a geometric sequence, we start with the general form:

\[
a_n = a \cdot r^{n-1}
\]

where:
- \( a \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the term number.

Given points:
- \( (1, 8) \), which tells us that when \( n = 1 \), \( a_1 = 8 \),
- \( (2, 4) \), which tells us that when \( n = 2 \), \( a_2 = 4 \).

### Step 1: Find the Common Ratio \( r \)

Since \( a_2 = a \cdot r = 4 \) and \( a_1 = a = 8 \), we can solve for \( r \):
\[
r = \frac{a_2}{a_1} = \frac{4}{8} = \frac{1}{2}
\]

### Step 2: Write the Explicit Formula

Now that we know \( a = 8 \) and \( r = \frac{1}{2} \), we can substitute these values into the general formula:

\[
a_n = 8 \cdot \left(\frac{1}{2}\right)^{n-1}
\]

### Final Answer

The explicit formula for the geometric sequence is:

\[
a_n = 8 \cdot \left(\frac{1}{2}\right)^{n-1}
\]

Would you like further explanation, or do you have any questions?

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Here are related questions to expand on this topic:
1. How do we find the sum of a finite geometric sequence?
2. What would the sequence look like if \( r = 2 \) instead?
3. How do we find the \( n \)-th term if only the first and third terms are given?
4. What happens to a geometric sequence if \( |r| > 1 \)?
5. How do you determine if a sequence is arithmetic or geometric?

**Tip:** In a geometric sequence, the common ratio \( r \) is always the factor by which each term is multiplied to get the next term.

write an explicit formula for a geometric sequence that has the points (1, 8) and (2, 4)

Explore how to write an explicit formula for a geometric sequence with given points (1, 8) and (2, 4). Learn key steps in finding the first term and common ratio, and applying them to the geometric sequence formula. Ideal for students in Grades 9-10.

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