UpStudy Free Solution:

To find the next number in the sequence \(5, 3, 9, 13, 23\), let's first look at the differences between consecutive terms:

- \(3 - 5 = - 2\)

- \(9 - 3 = 6\)

- \(13 - 9 = 4\)

- \(23 - 13 = 10\)

The differences between the terms are: \(- 2, 6, 4, 10\).

Now, let's look at the differences between these differences:

- \(6 - ( - 2) = 8\)

- \(4 - 6 = - 2\)

- \(10 - 4 = 6\)

The second-level differences are: \(8, - 2, 6\).

To see if there's a pattern, let's look at the differences between these second-level differences:

- \(- 2 - 8 = - 10\)

- \(6 - ( - 2) = 8\)

The third-level differences are: \(- 10, 8\).

It appears that there might be a repeating pattern in the differences. Given the differences we have:

- First-level differences: \(- 2, 6, 4, 10\)

- Second-level differences: \(8, - 2, 6\)

- Third-level differences: \(- 10, 8\)

If we assume the pattern continues, the next second-level difference should be \(- 2\) (since \(8, - 2, 6\) appears to be a repeating pattern).

Adding \(- 2\) to the last first-level difference (which is \(10\)):

\[10 + ( - 2) = 8\]

Now, add this to the last term in the sequence (which is \(23\)):

\[23 + 8 = 31\]

So, the next number in the sequence is \(31\).

Supplemental Knowledge

Understanding sequences involves recognizing patterns in the differences between consecutive terms. Here are some key concepts that can help:

1. Arithmetic Sequences:

- An arithmetic sequence has a constant difference between consecutive terms.

- Example: \(2, 5, 8, 11, \ldots \) (common difference = 3).

2. Geometric Sequences:

- A geometric sequence has a constant ratio between consecutive terms.

- Example: \(3, 6, 12, 24, \ldots \) (common ratio = 2).

3. Difference Sequences:

- For more complex sequences, examining the differences between terms can reveal patterns.

- First-level differences are the differences between consecutive terms of the original sequence.

- Second-level differences are the differences between consecutive first-level differences.

4. Higher-Level Differences:

- Sometimes it’s necessary to look at second or even third-level differences to identify a pattern.

- If higher-level differences form a repeating pattern or become constant, it can help predict future terms.

5. Example Analysis of Differences:

- Consider the sequence \(1, 4, 9, 16\).

- First-level differences: \(3, 5, 7\).

- Second-level differences: \(2\), which is constant.

Deciphering complex sequences can be both daunting and gratifying - it takes both practice and hard work! But once you do find hidden patterns, the reward can be immense! UpStudy offers assistance for this and many other mathematical topics!
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