1. Year 7 Sequences Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 23 rd July 2015 Objectives: Understand term-to-term vs position-to-term rules. Be able to generate terms of a sequence given a formula. Find the formula for a linear sequence. Be able to find a term of an oscillating sequence.

2. Teacher Guidance

3. STARTER :: What’s next in each sequence? A sequence is simply an ordered list of items (possibly infinitely long), usually with some kind of pattern. What are the next two terms in each sequence? ? ? ? ? ? ? ? ? Only 1 term needed. (Nicked off 2015’s ‘Child Genius’ on Channel 4) a b c d e f g h

4. Term-to-term rules Some sequences we can generated by stating a rule to say how to generate the next term given the previous term(s). DescriptionFirst 5 terms The first term of a sequence is 1. +3 to each term to get the next. 1, 4, 7, 10, 13 3, 6, 12, 24, 48 The first two terms are 0 and 1. Add the last two terms to get the next. 0, 1, 1, 2, 3 (known as the Fibonacci sequence) ? ? ? What might be the disadvantage of using a term-to-term rule? To get a particular term in the sequence, we have to generate all the terms in the sequence before it. This is rather slow if you say want to know the 1000 th term! ?

5. [JMC 2009 Q11] In a sequence of numbers, each term after the first three terms is the sum of the previous three terms. The first three terms are -3, 0, 2. Which is the first term to exceed 100? A 11 th termB 12 th termC 13 th term D 14 th termE 15 th term JMC Puzzle CBADE Terms are: -3, 0, 2, -1, 1, 2, 2, 5, 9, 16, 30, 55, 101

6. It’s sometimes more helpful to be able to generate a term of a formula based on its position in the sequence. We could use it to say find the 300 th term of a sequence without having to write all the terms out! 1 st term2 nd term3 rd term4 th term This formula gives the triangular numbers! ???? ???? ???? ???? ???? ????

7. Check Your Understanding ? ? ? ?

8. Challenge Your Neighbour A MERIT for the most interesting sequence!

9. Exercise 1 MOVE QUESTION TO LINEAR DIOPHANTINE EQS: [JMO 2008 B6] In a sequence of positive integers, each term is larger than the previous term. Also, after the first two terms, each term is the sum of the previous two terms. The eighth term of the sequence is 390. What is the ninth term? Solution: 631 (Hint: perhaps represent the first two terms algebraically?) ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 3 4 5 6 

10. ? Picture Sequence Puzzle… What are the next two pictures in this sequence?

11. Generating Sequences on your Calculator! (just for fun)

12. ? ? ? ? ? ? ?

13. ?? Bro Side Note: Why do you think this is known as a ‘linear’ sequence? If you plotted each position with the term on some axes (e.g. for this sequence (1,5),(2,9),(3,13),(4,17), …, it would form a straight line. The word ‘linear’ means ‘straight’. ?

14. More examples Quickfire Questions: 100 th term: ? ? ? ? ? ?? ?? ?? ??

15. Test Your Understanding 100 th term: ?? ?? ?? ??

16. Is a number in the sequence? ? ?

17. Exercise 2 2 1 3 4 5 6 a b c d e f g a b c d a b a b c d ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

18. Levelled Game

19. Oscillating Sequences There are many places in mathematics where sequences repeat. e.g. tiffintiffintiffintiffintiffintiffin… What is the 60 th letter in this sequence?‘n’ What is the 100 th letter in this sequence?‘f’ What is the 175 th letter in this sequence?‘t’ ? ? ?

20. Oscillating Sequences tiffintiffintiffintiffintiffintiffin… 1 2 3 4 5 6 | 7 8 9 10 11 12 The sequence repeats every 6 letters. Therefore, if we find the remainder when we divide the position by 6, we can just compare against the first few letters. ? ? ? Bro Mental Tip: 180 is clearly a multiple of 6, so we only have to divide 20 by 6 for the remainder.

21. Last Digits ? Consider the last digit when the power is 1, 2, 3, 4, … How does the sequence repeat?

22. Test Your Understanding A B ? ?

23. Exercise 3 1 2 3 4 5  a b c d a b c d e ? ? ? ? ? ?? ? ? ? ? ? ?

24. QQQ Time!