Insert 4 geometric means between the pair of numbers in parenthesis (3, 84)

To insert 4 geometric means between 3 and 84, we follow these steps:

### Step 1: Define the terms
Let the numbers in the geometric sequence be:
\[
3, G_1, G_2, G_3, G_4, 84
\]
where \( G_1, G_2, G_3, G_4 \) are the 4 geometric means we need to find.

### Step 2: Use the geometric sequence formula
In a geometric sequence, the ratio between consecutive terms is constant. Let the common ratio be \( r \). The general term of a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{n-1}
\]
where \( a_n \) is the nth term, and \( a_1 \) is the first term.

For this problem:
- First term \( a_1 = 3 \)
- Sixth term \( a_6 = 84 \) (since there are 5 terms between 3 and 84)

Thus:
\[
a_6 = a_1 \cdot r^5 = 84
\]
Substitute \( a_1 = 3 \):
\[
3 \cdot r^5 = 84
\]

### Step 3: Solve for the common ratio \( r \)
Divide both sides by 3:
\[
r^5 = \frac{84}{3} = 28
\]
Now solve for \( r \) by taking the 5th root of 28:
\[
r = \sqrt[5]{28}
\]

Using approximate values:
\[
r \approx 1.9037
\]

### Step 4: Find the geometric means
Now that we have the common ratio \( r \approx 1.9037 \), we can find the four geometric means by multiplying each successive term by \( r \).

1. \( G_1 = 3 \times 1.9037 \approx 5.711 \)
2. \( G_2 = G_1 \times 1.9037 \approx 10.871 \)
3. \( G_3 = G_2 \times 1.9037 \approx 20.690 \)
4. \( G_4 = G_3 \times 1.9037 \approx 39.407 \)

### Final Sequence:
\[
3, 5.711, 10.871, 20.690, 39.407, 84
\]

Would you like more details or have any questions about this?

### Related Questions:
1. How do you find the common ratio in a geometric sequence?
2. What is the difference between geometric and arithmetic sequences?
3. Can you insert more geometric means between any two numbers?
4. How do you find the nth term of a geometric sequence?
5. What are some applications of geometric sequences in real life?

### Tip:
When dealing with geometric sequences, always remember that the ratio between any two consecutive terms remains constant throughout the sequence!

Insert 4 geometric means between the pair of numbers in parenthesis (3, 84).

Discover how to insert 4 geometric means between the numbers 3 and 84. This guide covers the geometric sequence formula, solving for the common ratio, and finding the intermediate values. Suitable for high school students studying advanced algebra and sequences.

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