P1 Chapter 8 CIE Centre A-level Pure Maths © Adam Gibson

Sequences Look at the following lists of numbers – what is next? … … We cannot always deduce the sequence rule from the first few terms A “sequence” “terms” !

Defining a sequence or series – deductive and inductive The second sequence on the previous slide can be expressed with a simple formula: This formula allows you to deduce the value of for any r. However, sometimes this is not convenient and we should define the sequence inductively. Example: write the first 5 terms of the sequence Try different starting values; look for patterns

The prolific genius Gauss was born in Brunswick (Ger. Braunschweig), in the Duchy of Brunswick-Lüneburg as the only son of uneducated lower-class parents. According to legend, his gifts became apparent at the age of three when he corrected, in his head, an error his father had made on paper while calculating finances. Another story has it that in elementary school his teacher tried to occupy pupils by making them add up the integers from 1 to 100. The young Gauss produced the correct answer within seconds by a flash of mathematical insight, to the astonishment of all. Gauss had realized that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: = 101, = 101, = 101, and so on, for a total sum of 50 × 101 = 5050.

Important sequences The triangle sequence is the one discovered by the young Gauss: The factorial sequence is as follows: Can we give an inductive definition of this sequence? and is marked with ‘!’ Give an inductive definition. Then calculate : 5!, 6!, 1!, 0!, (32!/30!)

Pascal sequences are a key part of mathematics. They take this form: a is any integer We can write the terms of Pascal sequences as: Important sequences.. Contd. Write the whole Pascal sequence for a = 6. What is

You want to save money at the bank. At the beginning of 2006 you have $ in the account. Your salary is $2,000 per month. Your salary increases by $1,000 each year (not each month!) At the end of each year, you will add 5% of your total annual salary to your bank savings account. Ignoring interest payment, how much will be in the account: After 32 years? After 41 years? Arithmetic sequences

Arithmetic sequences have a common difference: …. has a common difference …. has a common difference -3 We write the common difference as d, the starting Value a, and the last value l Remember Gauss’ problem? ………………………….100 So:

Arithmetic sequences General formula is: (see page 122 of textbook)

The savings problem a = $1200 (not $150!) d = $50 (why? It’s 5% of the $1,000 salary increase) n = 32 l = $1,200+31x50=$2,750 S = 0.5x32x( )=$63,200 The balance in the account = S+$150 = $63,350

Pascal’s sequence – what does it mean? Pascal’s sequence tells us the number of ways we can choose a objects from a set of r objects by definition; all Pascal sequences start with 1 because there is one way you can choose zero objects from 4. because there are 4 ways you can choose 1 object from 4.

Look: Pascal’s sequence – what does it mean? because there are 6 ways you can choose 2 objects from 4 Let’s choose 2 6 ways – assuming order doesn’t matter

Think Pascal’s sequence and algebra Consider the expression It can be shown that the coefficients are the same as those in Pascal’s triangle. Can we use this to find the coefficient of in the expression ? Why?

Pascal’s sequence and factorials We can! But we first need to know that: See page 132 of your textbook. So the coefficient of in is 286. Because

The Binomial Theorem States that … where a and b are real numbers and n is a natural number. and wherecan be calculated using the formula: YOU MUST COMMIT THIS TO MEMORY! More succinctly,

Tasks Calculate these: Give the complete expansion of Calculate correct to 5 decimal places.

An aside: the Fibonacci sequence Gradient = ?

Why is the gradient Φ? The key concept is self-similarity Fractals This one is based on the “Fibonacci Spiral”

Why is the gradient Φ? … Contd. Fibonacci numbers very often occur in nature: 0 a b c By definition, In the limit, Let Solve the equation!

Examples – how to use the Binomial Theorem Give the value of to 5 decimal places Consider the expression A special case of the Binomial Theorem Let’s expand it out: Simplify: What can we say about the terms if x is small …?

Examples – how to use the Binomial Theorem If x is small, the terms in the sequence get rapidly (exponentially) smaller. 1 st term: 2 nd term: 3 rd term: 4 th term: 5 th term: What is the last term?

Examples – how to use the Binomial Theorem There is no need to calculate the rest of the expansion; the terms are very small. So, to 5 decimal places: To extend this idea to a slightly more difficult case, try Q21 p. 136