Sequences Objectives Grade D Write the terms of

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. 3 n + 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. 2 n the coefficient is 2

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: 5, 7, 9, 11, . . . The dots show that the sequence continues 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, . . . +4 2) 0, 5, 10, 15, . . . +5 3) 16, 13, 10, 7, . . . -3 4) 3, 1, - 3, . . . -2 5) 1, 2, 4, 8, 16, x 2 6) 2, 3. 5, 5, 6. 5, . . . +1. 5 7) 0. 01, 0. 1, 1, 10, . . . x 10

Sequences To write the terms of a sequence given the nth term Given the expression: 2 n + 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 5 7 2 x 3+3 2 x 4+3 9 11 2 x 5+3 13

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

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 2 n +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 2 n +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 4 n 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: 4 n - 1

Sequences To summarise finding the nth term: • Find the term to term rule • Find how much you need to add or subtract to get the correct sequence • Check your nth term works for the 2 nd and 3 rd terms Try this: Find the nth term for the following sequence: +3 Write: 5, 8, 11, 14 3 n use n = 1 for the 1 st term would be 3 x 1 = 3 We need to + 2 to make the correct first term of 5 The nth term becomes 3 n + 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 n + 2 2) 1, 6, 11, 16, . . . 5 n - 4 3) 3, 10, 17, 24, . . . 7 n - 4 4) 10, 19, 28, 37, . . . 9 n + 1 5) 13, 10, 7, 4, . . . -3 n + 16 6) 20, 14, 8, 2, . . . -6 n + 26

Sequences Find the nth term of a series of diagrams Here is a series of diagrams 5 13 9 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 4 n + 1 17

Sequences Now try these:

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