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.