Sequences Objectives: Grade D: Write the terms of a sequence given the nth term Grade C: Find the nth term of a sequence or a series of diagrams

Sequences Definitions: Sequence A list of numbers or diagrams that are connected in some way. Term A term is a number and / or variable(s) connected with x and / or ÷ separated from anther term by an ‘+’ or ‘-’ operation. nth term The ‘general’ term used to describe a sequence, e.g. 3n + 1 If you are given the nth term you can find the terms of a sequence. Coefficient The number preceding a letter – the number that is used to multiply a letter. e.g. 2n the coefficient is 2

The term to term rule is the rule that connects numbers in a sequence: Sequences The term to term rule is the rule that connects numbers in a sequence: e.g. Find the term to term rule for this sequence: The dots show that the sequence continues 5, 7, 9, 11, ... Each term (consecutive number in the sequence) is 2 more than the one before it, so the rule is: +2

Sequences Now try these: Find the term to term rule for these sequences: 1) 3, 7, 11, 15, ... 2) 0, 5, 10, 15, ... 3) 16, 13, 10, 7, ... 4) 3, 1, -1, - 3, ... 5) 1, 2, 4, 8, 16, 6) 2, 3.5, 5, 6.5, ... 7) 0.01, 0.1, 1, 10, ... +4 +5 -3 -2 x 2 +1.5 x 10

Sequences To write the terms of a sequence given the nth term Given the expression: 2n + 3 write the first 5 terms In this expression the letter n represents the term number and thus if we substitute the term number for the letter n we will find value that particular term. The first 5 terms of the sequence will be using values for n of 1, 2, 3, 4 and 5 term 1 term 2 term 3 term 4 term 5 2 x 1 + 3 2 x 2 + 3 2 x 3 + 3 2 x 4 + 3 2 x 5 + 3 5 7 9 11 13

Sequences Now try these: Write the first 5 terms of these sequences: 1) n + 2 2) 2n + 5 3) 3n - 2 4) 5n + 3 5) -4n + 10 6) n2 + 2 7) 3n2 3, 4, 5, 6, 7 7, 9, 11, 13, 15 1, 4, 7, 10, 13 8, 13, 18, 23, 28 6, 2, - 2, - 6, - 10 3, 6, 11, 18, 27 3, 12, 27, 48, 75

n +2 and the sequence it generates: Sequences Look at the nth term n +2 and the sequence it generates: 3, 4, 5, 6, 7 Find the term to term rule: +1 Look at the nth term 2n +5 and the sequence it generates: 7, 9, 11, 13, 15 Find the term to term rule: +2 You will notice that the coefficient in each expression is the term to term rule: n +2 rule +1 2n +5 rule +2 The coefficient of n is 1 because if we multiply n by 1 it is still n

Sequences To find the nth term of a sequence or a series of diagrams this shows us that the coefficient for a sequence is the term to term rule, so we always find this first. Example: Find the nth term for the following sequence: 3, 7, 11, 15 The term to term rule is: +4 Therefore the coefficient of n is 4 so we write 4n However, if we find the first 4 terms of this we get: 4, 8, 12, 16 This doesn’t give the correct sequence, but you will notice that If you subtract 1 from each of these terms you get the correct sequence The full nth term is therefore: 4n - 1

Sequences To summarise finding the nth term: Find the term to term rule Find the term to term rule Find how much you need to add or subtract to get the correct sequence Find how much you need to add or subtract to get the correct sequence Check your nth term works for the 2nd and 3rd terms Try this: Find the nth term for the following sequence: 5, 8, 11, 14 +3 Write: 3n use n = 1 for the 1st term the 1st term would be 3 x 1 = 3 We need to + 2 to make the correct first term of 5 The nth term becomes 3n + 2 Check this in terms 2 and 3 3 x 2 + 2 = 8 3 x 3 + 2 = 11

Sequences Now try these: Find the nth term for these sequences: 1) 4, 6, 8, 10, ... 2) 1, 6, 11, 16, ... 3) 3, 10, 17, 24, ... 4) 10, 19, 28, 37, ... 5) 13, 10, 7, 4, ... 20, 14, 8, 2, ... 2n + 2 5n - 4 7n - 4 9n + 1 -3n + 16 -6n + 26

Sequences Find the nth term of a series of diagrams Here is a series of diagrams 5 9 13 17 Write the number of matches in each pattern Now we have a sequence of numbers from which we can find the nth term The term to term rule is +4 The nth term is therefore 4n + 1

Sequences Now try these:

Sequences Now try these: 4 7 13 10 3n + 1 6 10 14 18 4n + 2 16 28 40 12n + 4