https://file.aiquickdraw.com/mathbot/file/1727805346220aoczuwyr.jpg

Let's analyze and solve the Fibonacci-type sequences provided in the image:

### a) \( 1, 5, 6, 11, 17, \dots \)

This is a Fibonacci-type sequence, which means each term after the first two is the sum of the two preceding terms.

- First, identify the recurrence relation:
  \[
  1 + 5 = 6
  \]
  \[
  5 + 6 = 11
  \]
  \[
  6 + 11 = 17
  \]
  So, continuing this pattern:
  \[
  11 + 17 = 28
  \]
  \[
  17 + 28 = 45
  \]

The next two terms for part (a) are **28** and **45**.

### b) \( 2, 5, 7, 12, 19, \dots \)

Again, it's a Fibonacci-type sequence, so each term is the sum of the two preceding terms.

- Recurrence relation:
  \[
  2 + 5 = 7
  \]
  \[
  5 + 7 = 12
  \]
  \[
  7 + 12 = 19
  \]
  Continuing the pattern:
  \[
  12 + 19 = 31
  \]
  \[
  19 + 31 = 50
  \]

The next two terms for part (b) are **31** and **50**.

### Final answers:
- Part (a): 28, 45
- Part (b): 31, 50

Would you like more details or have any questions?

Here are five related questions to deepen your understanding:
1. What is the general formula for a Fibonacci-type sequence?
2. How do Fibonacci-type sequences differ from standard arithmetic or geometric sequences?
3. Can you derive a recursive formula for the given sequences?
4. What would happen if the first two terms were different? How would that affect the sequence?
5. Can Fibonacci-type sequences be applied to real-world problems?

**Tip:** Fibonacci-type sequences are examples of recurrence relations, a key concept in many fields of mathematics and computer science.

Calculate the next two terms in each of these Fibonacci-type sequences:

a) 1, 5, 6, 11, 17, ...
b) 2, 5, 7, 12, 19, ...

Solve the Fibonacci-type sequences and calculate the next two terms in the sequences a) 1, 5, 6, 11, 17, ... and b) 2, 5, 7, 12, 19, .... Learn the recurrence relation method and key mathematical concepts behind Fibonacci sequences. Suitable for Grades 6-8.

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