Numbers

Natural Number: any whole number greater than zero

Integer: a positive or negative whole number or zero.

Prime Number:a whole number that can only be divided without a remainder by itself and on

Common Factor: a number that two or more other numbers can be divided by exactly. For example, 4 is a common divisor of 8, 12, and 20

Common Multiple: a number that can be divided exactly by two or more other numbers. For example, 12 is a common multiple of 2, 3, and 4.

Rational Number: a whole number or the quotient of any whole numbers, excluding zero as a denominator

Irrational Number: any real number that cannot be expressed as the exact ratio of two integers, e.g. √2 and π.

Sequences A number sequence is a set of numbers (terms) in which a pattern can be seen and a rule can be used to find every term in the sequence.For example:   5, 10, 20, 40… double the last term each time…..80, 1603, 5, 7, 9,…..Add two each time…..11, 13 25, 21, 17, 13,…minus four each time…..9, 5   The above examples have simple patterns, with harder sequences we need to look for a pattern and then establish the rule in order to calculate any term in the sequence. nth Term The rule for finding any term is called the nth term. For example: Given the sequence 6, 10, 14, 18,……     a) Find the nth term; b) Find the 20th term; c) If the nth term is 42, what is the value ofn?   We look at the differences between each term:     6 10 14 18 \ _/\_ /\_ / 4 4 4 The difference is four.       The general formula for the nth term is:     nth term = a + (n – 1)d where a = the first term = 6, n = the number of the term, and d = the difference = 4.     a) For this sequence: nth term = 6 + (n – 1) 4 = 6 + 4n – 4 = 2 + 4n       Now we'll use this formula to work out the value of any term in the sequence.       b) 20th term = 2 + 4 x 20 = 82 because n = 20       c) nth term = 42     42 = 2 + 4n 40 = 4n n = 10     So the 10th term is 42. The above formula will work for any linear sequence. In a linear sequence, the difference is constant. 4 in the sequence above. Quadratic Sequences   In this type, the first difference is not constant. The second difference gives a constant. For example: 3, 8, 15, 24, 36……… is a sequence.   3 8 15 24 35 \_/\_ /\_ /\_ / 5 7 9 11 \_ /\_ /\_ / 2 2 2 (first difference) (second difference) This is a quadratic sequence as the second difference is a constant (in this case, 2). The general formula for a quadratic is: nth term = a + (n – 1)d1 + ½(n – 1)(n – 2)d2 Where a = first term d1= first difference d2= second difference = 3 = 5 = 2     nth term = 3 + (n – 1)5 + ½ (n – 1)(n – 2)2 = 3 + 5n – 5 + n2 – 3n + 2 = 2n + n2 We can use this to find the 100th term:     100th term = 1 002 + 200 = 10 200 If second difference is a constant we can use this method for any of the quadratic sequence.