1. Source: http://nrich.maths.org/475 Loopy
A Fractional Sequence
Source:
Loopy

2. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =3, 𝑎 2 =7 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?

3. 𝑎 3 = 1+𝑏 𝑎 A Fractional Sequence 𝒌 𝒂 𝒌2 𝑏 3 4 5 6 7 8 9 10 11 12
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏
𝑎 3 = 1+𝑏 𝑎

4. 𝑎 4 = 1+ 1+𝑏 𝑎 𝑏 = 1+𝑎+𝑏 𝑎𝑏 A Fractional Sequence 𝒌 𝒂 𝒌2 𝑏 3
1+𝑏 𝑎 4 5 6 7 8 9 10 11 12
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏
𝑎 4 = 𝑏 𝑎 𝑏
= 1+𝑎+𝑏 𝑎𝑏

5. 𝑎 5 = 1+ 1+𝑎+𝑏 𝑎𝑏 1+𝑏 𝑎 = 𝑎𝑏+1+𝑎+𝑏 𝑎𝑏 1+𝑏 𝑎 = 1+𝑎 1+𝑏 𝑏 1+𝑏 = 1+𝑎 𝑏
A Fractional Sequence 𝒌
𝒂 𝒌 1 𝑎 2 𝑏 3
1+𝑏 𝑎 4
1+𝑎+𝑏 𝑎𝑏 5 6 7 8 9 10 11 12
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏
𝑎 5 = 𝑎+𝑏 𝑎𝑏 𝑏 𝑎
= 𝑎𝑏+1+𝑎+𝑏 𝑎𝑏 𝑏 𝑎
= 1+𝑎 1+𝑏 𝑏 1+𝑏 = 1+𝑎 𝑏

6. 𝑎 6 = 1+ 1+𝑎 𝑏 1+𝑎+𝑏 𝑎𝑏 = 1+𝑎+𝑏 𝑏 1+𝑎+𝑏 𝑎𝑏 =𝑎 A Fractional Sequence 𝒌
𝒂 𝒌 1 𝑎 2 𝑏 3
1+𝑏 𝑎 4
1+𝑎+𝑏 𝑎𝑏 5
1+𝑎 𝑏 6 7 8 9 10 11 12
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏
𝑎 6 = 𝑎 𝑏 𝑎+𝑏 𝑎𝑏
= 1+𝑎+𝑏 𝑏 𝑎+𝑏 𝑎𝑏 =𝑎

7. 𝑎 7 = 1+𝑎 1+𝑎 𝑏 =𝑏 A Fractional Sequence 𝒌 𝒂 𝒌2 𝑏 3
1+𝑏 𝑎 4
1+𝑎+𝑏 𝑎𝑏 5
1+𝑎 𝑏 6 7 8 9 10 11 12
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏
𝑎 7 = 1+𝑎 1+𝑎 𝑏 =𝑏

8. A Fractional Sequence 𝒌 𝒂 𝒌 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏3
1+𝑏 𝑎 4
1+𝑎+𝑏 𝑎𝑏 5
1+𝑎 𝑏 6 7 8 9 10 11 12
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏
The sequence loops and so we can go straight to the 11th and 12th terms.

9. A Fractional Sequence 𝒌 𝒂 𝒌 𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏3
1+𝑏 𝑎 4
1+𝑎+𝑏 𝑎𝑏 5
1+𝑎 𝑏 6 7 8 9 10 11 𝒂 12 𝒃
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 Let 𝑎 1 =𝑎, 𝑎 2 =𝑏
The sequence loops and so we can go straight to the 11th and 12th terms.

11. RESOURCES

12. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =3, 𝑎 2 =7 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

13. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =4, 𝑎 2 =3 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

14. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =2, 𝑎 2 =5 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

15. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =11, 𝑎 2 =17 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

16. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =13, 𝑎 2 =8 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

17. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =17, 𝑎 2 =5 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

18. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =17, 𝑎 2 =3 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

19. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =333, 𝑎 2 =297 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

20. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =997, 𝑎 2 =21 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

21. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =9, 𝑎 2 =3 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

22. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =−2, 𝑎 2 =5 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

23. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =3, 𝑎 2 =−3 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

24. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =100, 𝑎 2 =7 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

25. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =1000, 𝑎 2 =7 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

26. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =1000, 𝑎 2 =351 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23

27. A Fractional Sequence
The terms of a sequence are given by the inductive formula:
𝑎 𝑘 = 1+ 𝑎 𝑘−1 𝑎 𝑘−2 with 𝑎 1 =537, 𝑎 2 =11 𝒌
𝒂 𝒌 1 2 3 4 5 6 7 8 9 10 11 12
Find the values of 𝑎 11 and 𝑎 12 .
Experiment with your own values of 𝑎 1 and 𝑎 2
Make a conjecture about the behaviour of this sequence.
Can you prove your conjecture?
SIC_23