https://file.aiquickdraw.com/mathbot/file/1725366365573l0lhhodi.jpg

To solve the given problem, we need to determine the patterns in each of the sequences provided in the image. Let's analyze them step-by-step.

### Sequence 1: \(1, 4, 9, 16, 25, \ldots\)

This sequence appears to be the sequence of perfect squares:
\[
1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16, \quad 5^2 = 25, \ldots
\]

The next terms in this sequence would be \(6^2\) and \(7^2\):
\[
6^2 = 36, \quad 7^2 = 49.
\]

So, the next two numbers are **36, 49**.

### Sequence 2: \(8, 3, 9, 4, 10, 5, \ldots\)

This sequence alternates between two patterns:
1. Increasing numbers starting from 8: \(8, 9, 10, \ldots\)
2. Increasing numbers starting from 3: \(3, 4, 5, \ldots\)

The next two numbers would be \(11\) (next in the 8, 9, 10... pattern) and \(6\) (next in the 3, 4, 5... pattern).

So, the next two numbers are **11, 6**.

### Sequence 3: \(1, 1, 2, 3, 5, 8, \ldots\)

This sequence follows the Fibonacci pattern, where each number is the sum of the two preceding ones:
\[
1, \quad 1, \quad 2 \,(1+1), \quad 3 \,(1+2), \quad 5 \,(2+3), \quad 8 \,(3+5), \ldots
\]

The next terms would be:
\[
5 + 8 = 13, \quad 8 + 13 = 21.
\]

So, the next two numbers are **13, 21**.

### Sequence 4: \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots\)

This sequence follows a pattern where each term is \(\frac{n}{n+1}\):
\[
\frac{1}{2}, \quad \frac{2}{3}, \quad \frac{3}{4}, \quad \frac{4}{5}, \quad \frac{5}{6}, \ldots
\]

The next terms are:
\[
\frac{6}{7}, \quad \frac{7}{8}.
\]

So, the next two fractions are **\(\frac{6}{7}, \frac{7}{8}\)**.

### Summary of Next Numbers

1. **36, 49**
2. **11, 6**
3. **13, 21**
4. **\(\frac{6}{7}, \frac{7}{8}\)**

Would you like further details or explanations on any of the sequences? 

### Related Questions:

1. What is the general formula for the \(n\)-th term of a perfect square sequence?
2. How can alternating sequences like in the second example be identified in different contexts?
3. What are other examples of mathematical sequences with alternating patterns?
4. How does the Fibonacci sequence relate to the golden ratio?
5. What are some applications of the Fibonacci sequence in real-world scenarios?

### Tip:

When identifying patterns in sequences, look for common operations (addition, multiplication) or properties (like perfect squares or Fibonacci numbers) to determine the rule governing the sequence.

Explore detailed analysis of mathematical sequences including perfect squares, Fibonacci numbers, and fractions. Understand patterns and rules governing each sequence with examples and solutions. Ideal for advanced mathematics enthusiasts.

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