The document discusses various types of number sequences and recurrence relations. It defines key terms like consecutive terms, linear recurrence relations, convergence, divergence and limits of sequences. Examples are provided to illustrate recurrence relations, calculating sequence terms using a calculator, solving recurrence relations, linked recurrence relations, and special sequences like arithmetic, geometric and Fibonacci sequences.

1. Higher Maths 1 4 Sequences UNIT OUTCOME SLIDE

2. Higher Maths 1 4 Sequences Number Sequences UNIT OUTCOME NOTE The consecutive numbers or terms in any sequence can be written as SLIDE u 0 , u 1 , u 2 , u 3 , u 4 , u 5 ... u n consecutive means ‘ next to each other’ u n +1 u n the n th number in a sequence the number after the n th number (the ‘next’ number) u n –1 the number before the n th number (the ‘previous’ number)

3. Higher Maths 1 4 Sequences Basic Recurrence Relations UNIT OUTCOME NOTE It is often possible to write the next number in a sequence as a function of the previous numbers. SLIDE u n = 2 u n –1 – 3 with u 0 = 4 4 , 5 , 7 , 11 , 19 , 35 , 67 … u n = a u n –1 + b or alternatively u n –1 × 2 – 3 u n The number sequence is as follows: is called a linear recurrence relation . u n +1 = a u n + b In number sequences, Example

4. Higher Maths 1 4 Sequences Sequences on a Calculator UNIT OUTCOME NOTE It is possible to use a scientific calculator to quickly generate consecutive numbers in a sequence. SLIDE Set up u 0 Example u n +1 = u n + 1 4 3 ( ) 2 – = Input the recurrence relation, using the answer button as u n with Step 1: Step 2: u 0 = -2 Repeatedly press equals to generate the number sequence. Step 3: 4 ÷ 3 × ANS + 1

5. Higher Maths 1 4 Sequences Convergence and Divergence. UNIT OUTCOME NOTE Some sequences never stop increasing, while others eventually settle at a particular number. SLIDE If the numbers in a sequence continue to get further and further apart, the sequence diverges . u n n u n n If a sequence tends towards a limit , it is described as convergent . u n n L L

6. Higher Maths 1 4 Sequences Limit of a Sequence UNIT OUTCOME NOTE For any linear recurrence relation SLIDE u n +1 = a u n + b -1 < a < 1 If it exists, the limit of a convergent sequence can be calculated from a simple formula. L = b 1 – a the sequence tends to a limit L if Limit of a Linear Recurrence Relation Convergence or divergence is not affected by u 0

7. Higher Maths 1 4 Sequences Solving Recurrence Relations UNIT OUTCOME NOTE It is possible to find missing values in a recurrence relation by using consecutive terms to form simultaneous equations. SLIDE Example A linear recurrence relation has terms u n +1 = a u n + b u 1 = 5 , u 2 = 9.5 and u 3 = 20.75 . Find the values of a and b . 9.5 = 5 a + b u 2 = a u 1 + b 20.75 = 9.5 a + b u 3 = a u 2 + b Solving equations and simultaneously gives u n +1 = a u n + b and u n +1 = 2.5 u n – 3 a = 2.5 and b = - 3 2 1 1 2

8. Higher Maths 1 4 Sequences Linked Recurrence Relations UNIT OUTCOME NOTE SLIDE Example Two libraries, A and B, have 100 books in total. a n + b n = 100 a n +1 = 0.85 a n + 0.45 b n b n = 100 – a n = 0.85 a n + 0.45 ( 100 – a n ) = 0.4 a n + 45 Every week, 85 % of the books borrowed from library A are returned there, while the other 15 % are returned to library B instead. Only 55 % of the books borrowed from library B are returned there, with the other 45 % returned to library A. Investigate what happens. L a = b 1 – a = 45 1 – 0.4 = 75 L b = 25

9. Higher Maths 1 4 Sequences Special Sequences UNIT OUTCOME NOTE There are some important sequences which it is useful to know. SLIDE u n +1 = a u n u n +1 = u n + b u n +2 = u n + u n +1 Sequence Recurrence Relation Example Arithmetic Geometric Fibonacci 2, 8, 14, 20, 26, 32 ... 3, 12, 48, 192, 768 ... with u 0 = 0 and u 1 = 1 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ... discovered 1202