Hi where did you get the r³ in problem 3 which is 56 = 7r³

Great question! Let's break down where the equation \(56 = 7r^3\) comes from in Problem 3.

### Problem Recap:
You were given the sequence \(7, \_, \_, 56\) and asked to find the two missing terms.

### Understanding the Sequence:
In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio \(r\). If the sequence is \(a_1, a_2, a_3, a_4\), then:
- \(a_2 = a_1 \times r\)
- \(a_3 = a_2 \times r = a_1 \times r^2\)
- \(a_4 = a_3 \times r = a_1 \times r^3\)

### Setting Up the Equation:
Here, the first term \(a_1 = 7\), and the fourth term \(a_4 = 56\). We want to relate the first term to the fourth term using the common ratio \(r\).

- The fourth term is expressed as:
  \[
  a_4 = a_1 \times r^3
  \]
  Substituting the values we know:
  \[
  56 = 7 \times r^3
  \]

This equation \(56 = 7r^3\) comes from the relationship between the first term and the fourth term in a geometric sequence. Solving for \(r\) allows us to find the common ratio and subsequently the missing terms.

### Solving for \(r\):
\[
r^3 = \frac{56}{7} = 8
\]
\[
r = \sqrt[3]{8} = 2
\]

With \(r = 2\), you can then find the missing terms as:
- \(x_1 = 7 \times 2 = 14\)
- \(x_2 = 14 \times 2 = 28\)

So the full sequence is \(7, 14, 28, 56\).

Would you like to explore more on this, or do you have other questions? Here are some questions related to geometric sequences:

1. How do you solve for the common ratio when multiple terms are missing?
2. Can you find the general term formula for a geometric sequence?
3. How do geometric sequences differ from arithmetic sequences in terms of progression?
4. What is the importance of the common ratio in determining the behavior of the sequence?
5. How would you handle a geometric sequence with non-integer or fractional terms?

**Tip:** Always express each term in a sequence using its relationship with the first term and the common ratio to solve for missing terms accurately.

Explore how to solve for missing terms in a geometric sequence with a detailed example. Learn about the relationship between terms using the common ratio and initial term. Suitable for high school students.

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