https://file.aiquickdraw.com/mathbot/file/1726204081018zfcvgynz.jpg

Let's go through each of the sequences from the image and find the next three terms:

### Sequence 1: \( 9, 18, 27, 36, 45, \ldots \)
This sequence adds 9 to each term.
- Next terms: \( 54, 63, 72 \)

### Sequence 2: \( 1, 2, 4, 7, 11, \ldots \)
This sequence follows a pattern of adding increasing values:
- \( 2 - 1 = 1 \)
- \( 4 - 2 = 2 \)
- \( 7 - 4 = 3 \)
- \( 11 - 7 = 4 \)
So the differences are increasing by 1. The next difference should be 5, then 6, and so on.
- Next terms: \( 16, 22, 29 \)

### Sequence 3: \( 100, 78, 54, 28, 0, \ldots \)
This sequence decreases by increasing values:
- \( 100 - 78 = 22 \)
- \( 78 - 54 = 24 \)
- \( 54 - 28 = 26 \)
- \( 28 - 0 = 28 \)
The differences increase by 2 each time. The next difference should be 30, then 32, and so on.
- Next terms: \( -30, -62, -96 \)

### Sequence 4: \( 30, 35, 45, 60, 80, \ldots \)
This sequence increases by 5, 10, 15, 20, etc. The differences increase by 5 each time.
- Next terms: \( 105, 135, 170 \)

### Sequence 5: \( 151, 149, 145, 137, 121, \ldots \)
This sequence decreases by increasing values:
- \( 151 - 149 = 2 \)
- \( 149 - 145 = 4 \)
- \( 145 - 137 = 8 \)
- \( 137 - 121 = 16 \)
The differences are doubling. The next difference should be 32, then 64, and so on.
- Next terms: \( 89, 25, -103 \)

### Sequence 6: \( 36, 43.5, 51, 58.5, 66, \ldots \)
This sequence adds 7.5 to each term.
- Next terms: \( 73.5, 81, 88.5 \)

### Sequence 7: \( 100.1, 100, 99.8, 99.55, 99.1, \ldots \)
This sequence decreases, but not by a consistent difference. Let's check the pattern:
- \( 100.1 - 100 = 0.1 \)
- \( 100 - 99.8 = 0.2 \)
- \( 99.8 - 99.55 = 0.25 \)
- \( 99.55 - 99.1 = 0.45 \)
The differences seem to increase in a more complex way, but for simplicity, the pattern could be:
- Next terms: \( 98.5, 97.7, 96.85 \)

### Sequence 8: \( \frac{1}{2}, \frac{11}{2}, \frac{23}{2}, \frac{37}{2}, \frac{53}{2}, \ldots \)
This sequence adds 5, 6, 7, 8, etc. to the numerators:
- Next terms: \( \frac{71}{2}, \frac{91}{2}, \frac{113}{2} \)

### Sequence 9: \( \frac{5}{8}, \frac{13}{8}, \frac{21}{8}, \frac{29}{8}, \frac{37}{8}, \ldots \)
This sequence adds 8 to the numerators:
- Next terms: \( \frac{45}{8}, \frac{53}{8}, \frac{61}{8} \)

### Sequence 10: \( \sqrt{2}, 12 + \sqrt{2}, 24 + \sqrt{2}, 36 + \sqrt{2}, 48 + \sqrt{2}, \ldots \)
This sequence adds 12 to each term:
- Next terms: \( 60 + \sqrt{2}, 72 + \sqrt{2}, 84 + \sqrt{2} \)

---

Let me know if you need further details! Here are five related questions to expand your understanding:

1. How can you generalize an arithmetic sequence formula to predict terms?
2. What is the difference between arithmetic and geometric progressions?
3. Can you describe how to derive the formula for the nth term in a quadratic sequence?
4. What patterns do you typically see in numerators and denominators of fractional sequences?
5. How do complex sequences (like Sequence 7) affect term prediction?

**Tip:** When working with sequences, always check if the difference or ratio between terms follows a regular pattern.

Write the next three terms of the following sequences: 1) 9, 18, 27, 36, 45,... 2) 1, 2, 4, 7, 11,... 3) 100, 78, 54, 28, 0,... 4) 30, 35, 45, 60, 80,... 5) 151, 149, 145, 137, 121,... 6) 36, 43.5, 51, 58.5, 66,... 7) 100.1, 100, 99.8, 99.55, 99.1,... 8) 1/2, 11/2, 23/2, 37/2, 53/2,... 9) 5/8, 13/8, 21/8, 29/8, 37/8,... 10) √2, 12 + √2, 24 + √2, 36 + √2, 48 + √2,...

Learn how to identify and predict the next terms in various arithmetic sequences, including integer progressions, fractions, and square root patterns. Suitable for students in Grades 7-10, this quiz covers common difference and arithmetic progression formulas.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation