Describe the pattern of matchstick houses, calculate the number of matchsticks needed for different diagrams, and find the rule for this pattern.

The pattern shows houses made of matchsticks. Each subsequent diagram is formed by adding one more house to the right of the previous one, sharing a vertical matchstick with the house to its left.

To make the next diagram (Diagram 4), you must add one more house to the right of Diagram 3, sharing a vertical matchstick. Diagram 4 will consist of four houses joined together in a row.

• Diagram 1: 5 matchsticks
• Diagram 2: 5 (for 1st house) + 4 (for 2nd house) = 9 matchsticks
• Diagram 3: 9 (for 2 houses) + 4 (for 3rd house) = 13 matchsticks
• Diagram 4: 13 (for 3 houses) + 4 (for 4th house) = 17 matchsticks

The number of matchsticks increases by 4 for each additional house. This is an arithmetic sequence with a common difference of 4.
Let $n$ be the diagram number.
The rule is:

Diagram number $\rightarrow$ Multiply by 4 and add 1 $\rightarrow$ number of matches.

• For Diagram 2 ($n=2$): Number of matchsticks $= 4(2) + 1 = 8 + 1 = 9$. This matches the count.
• For Diagram 3 ($n=3$): Number of matchsticks $= 4(3) + 1 = 12 + 1 = 13$. This matches the count.

• For Diagram 5 ($n=5$): Number of matchsticks $= 4(5) + 1 = 20 + 1 = 21$.
• For Diagram 7 ($n=7$): Number of matchsticks $= 4(7) + 1 = 28 + 1 = 29$.
• For Diagram 12 ($n=12$): Number of matchsticks $= 4(12) + 1 = 48 + 1 = 49$.

Updated table:

We use the rule: Number of matchsticks $= 4n + 1$.
Given 101 matchsticks:

Subtract 1 from both sides:

Divide by 4:

So, 25 houses are joined together.

Final updated table:

.4: Number Sequences

For each sequence, the next three numbers are provided, along with the rule.

1. $3; 6; 9; 12; \boxed{\text{15; 18; 21}}$ Rule: Add 3 to the previous number.

2. $0; 2; 4; 6; \boxed{\text{8; 10; 12}}$ Rule: Add 2 to the previous number.

3. $78; 67; 62; \boxed{\text{51; 46; 35}}$ Rule: Alternately subtract 11, then subtract 5 from the previous number. ($78 - 11 = 67$, $67 - 5 = 62$, $62 - 11 = 51$, $51 - 5 = 46$, $46 - 11 = 35$)