1. What is happening here? 1, 1, 2, 3, 5, 8 What is after 8? What is the 10 th number?

2. What is happening here? 1, 1, 2, 3, 5, 8 What is after 8? What is the 10 th number?

3. Welcome to Core 2 - Sequences Know what the “recursive definition” of a “sequence” are. Understand how to generate the terms of a sequence from recursive definition Be able to solve problems using simultaneous equations using the above.

4. Sequences You are expected to understand 3 types of sequences. Recursive, Arithmetic, and geometric. Keywords Sequence: a list of numbers e.g. 1, 2, 3, 4, … Series: a sum of numbers e.g. 1 + 2 + 3 + 4 + …

5. Terms and notation 1, 1, 2, 3, 5, 8, … First term in seq. 4 th term in seq. n th term in seq. Sometimes n is another letter, for example, r, u r. What ever n =, that’s the number term you want

6. Recursive Sequence 1, 2, 3, 4, 5, 6, … A recursive sequence is a rule that gives you the next term using the preceding term. You’ll be given the first term.

7. Recursive Sequence 1, 2, 3, 4, 5, 6, … What is happening? What are we doing each time to get to the next term? The next term is found… …using The current/preceding term… …and adding one Equation is called the recursive definition

8. Recursive Sequence 1, 1, 2, 3, 5, 8, … This is the Fibonacci sequence The next term is found… …using The current/preceding term… …and adding the previous term

9. Recursive Sequence Given that the first term, u 1, is 5, what are the first five terms in this sequence? 5, 15, 45, 135, 405

10. Recursive Sequence What is the recursive definition for this sequence? 28, 14, 7, 3.5, 1.75

11. Questions Six questions on back, have a go at them! Be sure to mark your work using the solutions

12. Simultaneous Equations Problems will occur that involve the use of simultaneous equations, but they will not mention it at all.

13. Simultaneous Equations It will be set up like this: The recursive definition of a function is u n+1 = pu n + q The first three terms are 6, -7, 19 Find the values of p and q. Unknowns in equation & first 3 terms This set up is common and requires simultaneous equations!

14. Simultaneous Equations u n+1 = pu n + q The first three terms are 6, -7, 19 Have a go and finish off question

15. Try this one out The recursive definition of a function is u n+1 = a + bu n The first three terms are 8, 10, 11… Find the values of a and b. Find u 4

16. Try this one out The recursive definition of a function is u n+1 = a + bu n The first three terms are 6, 11.5, 19.75… Find the values of a and b. Find u 4

17. Try this one out The recursive definition of a function is u n = pu n-1 + q The first three terms are 2, -14, 50… Find the values of p and q. Find u 4 & u 5

18. Independent Study Exercise 2A p244 (solutions p430)