SEQUENCES

A sequence is a list of numbers or diagrams that are connected by a rule. 15 17 19 21 … and are sequences.

The numbers in a sequence are called the terms of the sequence The nth term is used to describe a general term in a sequence. If the nth term = 3n + 5 1st term = 3 × 1 + 5 = 8 2nd term = 3 × 2 + 5 = 11 3rd term = 3 × 3 + 5 = 14 4th term = 3 × 4 + 5 = 17 The sequence is 8, 11, 14, 17, … It is called a linear sequence because the differences between terms are all the same. The rule to find the next term (the term-to-term rule) is +3 or add 3.

1 Find the nth term of the linear sequence 10, 17, 24, 31, … Examples 1 Find the nth term of the linear sequence 10, 17, 24, 31, … Term position 1 2 3 4 The rule is multiply by 7 and then add 3. 10 17 24 31 + 7 + 7 + 7 nth term = 7n + 3

2 Find the nth term of the linear sequence 2, 5, 8, 11, … Examples 2 Find the nth term of the linear sequence 2, 5, 8, 11, … Term position 1 2 3 4 The rule is multiply by 3 and then take 1. 2 5 8 11 + 3 + 3 + 3 nth term = 3n - 1

3 Find the nth term of the linear sequence 9, 13, 17, 21, … Examples 3 Find the nth term of the linear sequence 9, 13, 17, 21, … Term position 1 2 3 4 The rule is multiply by 4 and then add 5. 9 13 17 21 + 4 + 4 + 4 nth term = 4n + 5

4 Find the nth term of the linear sequence 29, 22, 15, 8, … Examples 4 Find the nth term of the linear sequence 29, 22, 15, 8, … Term position 1 2 3 4 The rule is multiply by −7 and then add 36. 29 22 15 8 −7 −7 −7 nth term = 36 − 7n

b How many circles are in diagram 8? Examples 5 diagram 1 diagram 2 diagram 3 diagram 4 a Complete the table. Diagram 1 2 3 4 Number of circles × 4 4 8 12 16 + 4 + 4 + 4 Answer = 8 × 4 = 32 b How many circles are in diagram 8? c How many circles are in diagram n? Answer = 8 × n = 8n

b How many sticks are in diagram 27? Answer = 27 × 3 + 1 = 82 Examples 6 diagram 1 diagram 2 diagram 3 a Complete the table. Diagram 1 2 3 4 Number of sticks × 3 then +1 4 7 10 13 + 3 + 3 + 3 b How many sticks are in diagram 27? Answer = 27 × 3 + 1 = 82 c How many sticks are in diagram n? Answer = 3n + 1

Examples 7 nth term = 5n - 3. Find the term in the sequence that has a value of 122. add 3 to both sides divide both sides by 5 Answer: the 25th term

Examples 8 nth term = 6n + 15. Find the term in the sequence that has a value of 303. take 15 from both sides divide both sides by 6 Answer: the 48th term

Examples 9 nth term = 50 - 6n. Find the term in the sequence that has a value of -184. take 50 from both sides divide both sides by −6 Answer: the 39th term

Non linear sequences The sequence 5, 7, 10, 14, … is called a non-linear sequence because the difference between the terms is not the same. 5 7 10 14 + 2 + 3 + 4 You should be able to recognise and use the following non-linear sequences. Square numbers 1 4 9 16 nth term = n2

Cube numbers 1 8 27 64 nth term = n3 Powers For example 2, 4, 8, 16, … nth term = 2n 3, 9, 27, 81, … nth term = 3n 4, 16, 64, 256, … nth term = 4n

Examples 1 Find the nth term of the sequence 1, 3, 9, 27, … Term position 1 2 3 4 The rule involves powers of 3. 1 3 9 27 (= 30) (= 31) (= 32) (= 33) subtract 1 from the term position to find the power nth term =

Examples 2 Find the nth term of the sequence 4, 7, 12, 19, … Term position 1 2 3 4 The rule involves square numbers. 4 7 12 19 (= 12 + 3) (= 22 + 3) (= 32 + 3) (= 42 + 3) square the term position and add 3 nth term =

Examples 3 Find the nth term of the sequence 0, 7, 26, 63, … Term position 1 2 3 4 The rule involves cube numbers. 0 7 26 63 (= 13 − 1) (= 23 − 1) (= 33 − 1) (= 43 − 1) cube the term position and subtract 1 nth term =

Examples 4 The table shows the first four terms in three sequences A, B and C. Term 1 Term 2 Term 3 Term 4 Sequence A 3 6 11 18 Sequence B 9 27 81 Sequence C 16 63 a Find the nth term of sequence A. b Find the nth term of sequence B. c Find the nth term of sequence C. You need to notice that: sequence C = sequence B − sequence A

b How many circles are in diagram 20? Answer = Examples 5 diagram 1 diagram 2 diagram 3 diagram 4 a Complete the table. Diagram 1 2 3 4 5 Number of circles b How many circles are in diagram 20? Answer = c How many circles are in diagram n? Answer =

The diagram shows the growth of a plant over three years. Examples 6 Year 1 Year 2 Year 3 The diagram shows the growth of a plant over three years. Each year a flower is replaced by three stems and three flowers. a Complete the table. Year 1 2 3 4 5 Number of flowers 9 Number of stems 13 27 81 40 121

b How many flowers are there in year n? Examples Year 1 Year 2 Year 3 b How many flowers are there in year n? Using the answers from part a you can see a pattern. Year 1 2 3 4 5 Number of flowers 1 = 30 3 = 31 9 = 32 27= 33 81 = 34 So, number of flowers in year n =

c How many stems are there in year n? Examples Year 1 Year 2 Year 3 c How many stems are there in year n? Using the answers from part a you can see a pattern. Year 1 2 3 4 5 Number of stems So, number of stems in year n =

Further non-linear sequences Example 1 The nth term of the sequence 8, 16, 26, 38, … is given by the formula nth term = n2 + bn + c Find the values of b and c. The first term is 8, so substitute n = 1 into the formula The second term is 16, so substitute n = 2 into the formula Solving and simultaneously gives b = 5 and c = 2.

If you are asked to find the values of a, b and c when the nth term = an2 + bn + c then the problem is more complicated. One method is to look at the differences in the terms. For example 4 13 26 43 54 First difference + 9 + 13 + 17 + 21 Second difference + 4 + 4 + 4 The second differences are all the same. This means that it is a quadratic sequence A second difference of 2  the nth term will contain n2. A second difference of 4  the nth term will contain 2n2. A second difference of 6  the nth term will contain 3n2. So divide the second difference by 2 to find the coefficient of n2.

How many sticks are needed to make Examples 2 diagram 1 diagram 2 diagram 3 3 sticks 9 sticks 18 sticks How many sticks are needed to make a diagram 4 b diagram 5 c diagram n? a diagram 4 = 18 + 12 = 30 sticks b diagram 5 = 30 + 15 = 45 sticks

solving simultaneously Examples 2 diagram 1 diagram 2 diagram 3 3 sticks 9 sticks 18 sticks c 3 9 18 30 45 second difference is 3, so a = 3 ÷ 2 = 1.5 + 6 + 9 + 12 + 15 + 3 + 3 + 3 and simplifying and solving simultaneously or