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.