To solve these questions, let's follow a step-by-step approach.
### Part (b): Complete the Table
First, we observe the given pattern of the number of sticks for the pattern numbers:
| Pattern number | 1 | 2 | 3 | 4 | 5 |
|----------------|---|---|---|---|---|
| Number of sticks| 3 | 5 | 7 | 9 | 11|
From this table, you can observe that:
- When the pattern number is 1, the number of sticks is 3.
- When the pattern number is 2, the number of sticks is 5.
- When the pattern number is 3, the number of sticks is 7.
- When the pattern number is 4, the number of sticks is 9.
- When the pattern number is 5, the number of sticks is 11.
You can see that the difference between consecutive numbers of sticks is always 2 (5 - 3 = 2, 7 - 5 = 2, etc.), indicating that the sequence of the number of sticks is in arithmetic progression.
### Part (c): How many sticks make Pattern number 15?
To find the number of sticks for any pattern number in this arithmetic sequence, we use the formula for the n-th term of an arithmetic sequence:
[tex]\[a_n = a + (n - 1) \cdot d\][/tex]
Where:
- [tex]\(a_n\)[/tex] is the n-th term (number of sticks for the given pattern number)
- [tex]\(a\)[/tex] is the first term of the sequence
- [tex]\(d\)[/tex] is the common difference between the terms
- [tex]\(n\)[/tex] is the term number (pattern number)
From the table:
- The first term, [tex]\(a\)[/tex] = 3
- The common difference, [tex]\(d\)[/tex] = 2
- To find the number of sticks for pattern number 15, we set [tex]\(n = 15\)[/tex]
Now, substitute these values into the formula:
[tex]\[a_{15} = 3 + (15 - 1) \cdot 2\][/tex]
Calculate the terms step-by-step:
1. First, calculate [tex]\((n - 1)\)[/tex]:
[tex]\[15 - 1 = 14\][/tex]
2. Then multiply by the common difference [tex]\(d\)[/tex]:
[tex]\[14 \cdot 2 = 28\][/tex]
3. Finally, add this result to the first term [tex]\(a\)[/tex]:
[tex]\[3 + 28 = 31\][/tex]
Therefore, the number of sticks that make Pattern number 15 is [tex]\(31\)[/tex].
