Here are the questions from the PDF document titled "Patterns" that you uploaded:

1. Write the next three numbers in the sequence below:

   • 17, 22, 20, 25, 23, 28, 26, ...

2. Write the next three numbers in the sequence below:

   • 2, 6, 9, 27, 30, 90, 93, ...

3. Write the next two symbols in the sequence below:

   • ◀, ▷, ◁, ▶, ⊵, ◀, ▷, ...

4. Find the missing term in each pattern:

   • (a) 256, 128, 64, , 16, 8
   • (b) 7, 9, 12, 16, ...

5. Consider the following difference. Use your calculator to verify that the statements are true:

   • 62² − 52² = 11
   • 562² − 452² = 1111
   • 5562² − 4452² = 111,111
   • (a) Predict the next line in the sequence of differences and use your calculator to check your answer.
   • (b) What do you think the seventh line will be?

6. Determine if the following sequences are arithmetic, geometric, or neither:

   • (a) 8, 16, 24, 32, 40, ...
   • (b) 2, 8, 18, 32, 50, ...
   • (c) 4096, 1024, 256, 64, 16, ...
   • (d) 0, 1, 1, 2, 3, ...
   • (e) 5, 15, 45, 135, 405, ...

7. Find the next number in the sequence:

   • 3, 14, 31, 54, 83, 118, ...

8. The formula $S = \frac{n(n+1)}{2}$ can be used to find the sum of the first n natural numbers. Use the formula to find the sums of the following:

   • (a) $1 + 2 + 3 + \cdots + 22 + 23$
   • (b) $1 + 2 + 3 + \cdots + 117 + 118$

9. For the arithmetic sequence 1, 5, 9, 13, ... find the 23rd term.

10. For the geometric sequence 96, 48, 24, 12, 6, ... find the 10th term.

11. For an arithmetic sequence, the fifth term is 27 and the 11th term is 93. Using the formula $a_n = a_1 + d(n - 1)$, find the following:

    • (a) The common difference $d$.
    • (b) Use the common difference to find $a_1$.
    • (c) Use $a_1$ and $d$ to find $a_{39}$, the 39th term of the arithmetic sequence.

12. For the recursive formula $a_n = -2a_{n-1} + 1$, find the first six terms of the sequence using $a_1 = 2$.

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Would you like detailed solutions to any of these problems?

Here are 5 follow-up questions you could explore based on the above:

1. What type of sequence is represented by question 1, and how can we find the pattern?
2. What mathematical rule governs the changes in the second sequence (question 2)?
3. How do recursive sequences like the one in question 12 differ from arithmetic and geometric sequences?
4. What is the general formula for the nth term of an arithmetic sequence (referenced in question 11)?
5. How can you predict future differences in question 5 based on the given patterns?

Tip: When analyzing number patterns, look for consistent changes such as constant differences (arithmetic sequences) or ratios (geometric sequences) between terms.