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To solve the problem shown in the diagram, we need to find a rule that relates the number of honeycombs \( h \) to the pattern number \( n \).

### Observing the sequence:

1. For pattern \( n = 1 \), there is 1 honeycomb.
2. For pattern \( n = 2 \), there are 7 honeycombs (1 central honeycomb plus 6 surrounding it).
3. For pattern \( n = 3 \), there are 19 honeycombs.
4. For pattern \( n = 4 \), there are 37 honeycombs.
5. For pattern \( n = 5 \), there are 61 honeycombs.

### Looking for a pattern:
We can observe that the number of honeycombs increases in a quadratic way as \( n \) increases. This suggests a quadratic relationship of the form:

\[
h = an^2 + bn + c
\]

### Step 1: Set up a system of equations
Using the given values:

- For \( n = 1 \), \( h = 1 \):
  \[
  a(1)^2 + b(1) + c = 1 \implies a + b + c = 1
  \]

- For \( n = 2 \), \( h = 7 \):
  \[
  a(2)^2 + b(2) + c = 7 \implies 4a + 2b + c = 7
  \]

- For \( n = 3 \), \( h = 19 \):
  \[
  a(3)^2 + b(3) + c = 19 \implies 9a + 3b + c = 19
  \]

### Step 2: Solve the system of equations
We now solve this system of equations:

1. \( a + b + c = 1 \)
2. \( 4a + 2b + c = 7 \)
3. \( 9a + 3b + c = 19 \)

Subtract equation 1 from equation 2:
\[
(4a + 2b + c) - (a + b + c) = 7 - 1
\]
\[
3a + b = 6 \quad \text{(Equation 4)}
\]

Subtract equation 2 from equation 3:
\[
(9a + 3b + c) - (4a + 2b + c) = 19 - 7
\]
\[
5a + b = 12 \quad \text{(Equation 5)}
\]

Now subtract equation 4 from equation 5:
\[
(5a + b) - (3a + b) = 12 - 6
\]
\[
2a = 6 \implies a = 3
\]

Substitute \( a = 3 \) into equation 4:
\[
3(3) + b = 6 \implies 9 + b = 6 \implies b = -3
\]

Substitute \( a = 3 \) and \( b = -3 \) into equation 1:
\[
3 - 3 + c = 1 \implies c = 1
\]

### Step 3: Form the quadratic equation
Now that we have \( a = 3 \), \( b = -3 \), and \( c = 1 \), the rule that relates the number of honeycombs \( h \) to the pattern number \( n \) is:

\[
h = 3n^2 - 3n + 1
\]

### Verification:
- For \( n = 1 \): \( h = 3(1)^2 - 3(1) + 1 = 1 \) ✔️
- For \( n = 2 \): \( h = 3(2)^2 - 3(2) + 1 = 7 \) ✔️
- For \( n = 3 \): \( h = 3(3)^2 - 3(3) + 1 = 19 \) ✔️
- For \( n = 4 \): \( h = 3(4)^2 - 3(4) + 1 = 37 \) ✔️
- For \( n = 5 \): \( h = 3(5)^2 - 3(5) + 1 = 61 \) ✔️

The formula \( h = 3n^2 - 3n + 1 \) correctly models the sequence.

---

Do you want more details on any step or have any questions?

Here are 5 related questions to expand your understanding:
1. How can you generalize this approach to find patterns in other types of geometric sequences?
2. What would the next pattern, \( n = 6 \), look like and how many honeycombs would it have?
3. Can you derive a rule for sequences involving triangles or squares, using similar logic?
4. How does recognizing quadratic patterns help in solving real-world problems?
5. How would the equation change if each step added a different number of honeycombs?

**Tip:** When looking for patterns, always check if the differences between consecutive terms point to a specific type of function (linear, quadratic, etc.).

The diagram shows a sequence of expanding 'honeycombs'. Find the rule that relates the number of 'honeycombs' required (h) to the pattern number (n).

Learn how to determine the rule that relates the number of honeycombs (h) to the pattern number (n) in an expanding sequence. This detailed solution involves quadratic sequences, algebra, and pattern recognition. Perfect for high school students in Grades 9-12.

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