find the first term in the geometric sequence where a4 is equal to 4 and a7 is equal to 32

To find the first term in a geometric sequence given that \( a_4 = 4 \) and \( a_7 = 32 \), follow these steps:

### Step 1: Recall the formula for the \(n\)th term of a geometric sequence
The general formula for the \(n\)th term \(a_n\) of a geometric sequence is:
\[
a_n = a_1 \cdot r^{n-1}
\]
Where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

### Step 2: Write down the equations for \(a_4\) and \(a_7\)
Using the formula, we can write the equations for the 4th and 7th terms:
\[
a_4 = a_1 \cdot r^{4-1} = a_1 \cdot r^3 = 4
\]
\[
a_7 = a_1 \cdot r^{7-1} = a_1 \cdot r^6 = 32
\]

### Step 3: Divide the equations to eliminate \(a_1\)
Divide the equation for \(a_7\) by the equation for \(a_4\):
\[
\frac{a_7}{a_4} = \frac{a_1 \cdot r^6}{a_1 \cdot r^3} = \frac{32}{4}
\]
This simplifies to:
\[
r^3 = 8
\]
### Step 4: Solve for the common ratio \(r\)
\[
r = \sqrt[3]{8} = 2
\]

### Step 5: Solve for the first term \(a_1\)
Now that we know \(r = 2\), substitute it back into the equation for \(a_4\) to solve for \(a_1\):
\[
a_4 = a_1 \cdot r^3 = a_1 \cdot 2^3 = a_1 \cdot 8 = 4
\]
\[
a_1 = \frac{4}{8} = \frac{1}{2}
\]

### Final Answer:
The first term \(a_1\) of the geometric sequence is \(\frac{1}{2}\).

Would you like more details on any of these steps, or do you have any further questions?

Here are 5 questions related to the topic:
1. How do you find the common ratio when given two non-consecutive terms in a geometric sequence?
2. What is the effect of the common ratio on the growth or decay of a geometric sequence?
3. How can you find the sum of an infinite geometric series?
4. How do you determine the number of terms in a geometric sequence given the first term and the common ratio?
5. What happens when the common ratio of a geometric sequence is a negative number?

**Tip:** When dealing with geometric sequences, dividing one term by another term can help you find the common ratio quickly.

Learn how to find the first term in a geometric sequence given specific terms \( a_4 = 4 \) and \( a_7 = 32 \). This problem involves the application of geometric sequence formulas and is suitable for advanced high school students.

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