Identify the General Pattern

1.) 85, 79, 73, 67, 61,.....
2.) 15, 22, 29, 36,...
3.) 4, 8, 12, 16, 20,....
4.) 1, 5, 11, 19, 29
5.) 8, 19, 34, 53, ....

Let's analyze the general patterns for each sequence:

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**1.) 85, 79, 73, 67, 61,...**
- This is an **arithmetic sequence**, where each term decreases by 6.
  - Difference: \( 85 - 79 = 6 \), \( 79 - 73 = 6 \), and so on.
  - General form: \( a_n = 85 - 6(n-1) \)

**Pattern:** Decreasing by 6.

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**2.) 15, 22, 29, 36,...**
- This is an **arithmetic sequence**, where each term increases by 7.
  - Difference: \( 22 - 15 = 7 \), \( 29 - 22 = 7 \), and so on.
  - General form: \( a_n = 15 + 7(n-1) \)

**Pattern:** Increasing by 7.

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**3.) 4, 8, 12, 16, 20,...**
- This is an **arithmetic sequence**, where each term increases by 4.
  - Difference: \( 8 - 4 = 4 \), \( 12 - 8 = 4 \), and so on.
  - General form: \( a_n = 4 + 4(n-1) \)

**Pattern:** Increasing by 4.

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**4.) 1, 5, 11, 19, 29,...**
- This is a **quadratic sequence** where the second differences are constant.
  - First differences: \( 5 - 1 = 4 \), \( 11 - 5 = 6 \), \( 19 - 11 = 8 \), and so on.
  - Second differences: \( 6 - 4 = 2 \), \( 8 - 6 = 2 \), and so on.
  - General form (quadratic): \( a_n = n^2 + n - 1 \)

**Pattern:** Quadratic pattern with second difference constant.

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**5.) 8, 19, 34, 53,...**
- This is another **quadratic sequence**, as the second differences are constant.
  - First differences: \( 19 - 8 = 11 \), \( 34 - 19 = 15 \), \( 53 - 34 = 19 \).
  - Second differences: \( 15 - 11 = 4 \), \( 19 - 15 = 4 \).
  - General form (quadratic): \( a_n = 2n^2 + 5n + 1 \)

**Pattern:** Quadratic pattern with constant second differences.

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Would you like any further details or clarifications on any of these sequences?

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Here are 5 related questions:
1. What is the next term in the first sequence?
2. Can you generalize the formula for the second sequence?
3. How do you determine if a sequence is quadratic or arithmetic?
4. What are the first 10 terms of the third sequence?
5. How would you derive the quadratic form of the fourth sequence?

**Tip:** When analyzing sequences, always check for constant differences between terms; this helps to identify whether the sequence is arithmetic, geometric, or quadratic.

Explore how to identify patterns in arithmetic and quadratic sequences. Learn to analyze sequences like 85, 79, 73,... and 1, 5, 11,..., understanding the difference between arithmetic and quadratic sequences. This problem includes step-by-step solutions and is ideal for students in Grades 6-8.

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