KS3 Mathematics A4 Sequences of 27 © Boardworks Ltd 2004 Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts of 27 © Boardworks Ltd 2004 Introducing sequences In maths, we call a list of numbers in order a sequence Each number in a sequence is called a term 4, 8, 12, 16, 20, 24, 28, 32, 1st term 6th term If terms are next to each other they are referred to as consecutive terms When we write out sequences, consecutive terms are usually separated by commas of 27 © Boardworks Ltd 2004 Infinite and finite sequences A sequence can be infinite That means it continues forever For example, the sequence of multiples of 10, 10, 20 ,30, 40, 50, 60, 70, 80, 90 is infinite We show this by adding three dots at the end If a sequence has a fixed number of terms it is called a finite sequence For example, the sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite of 27 © Boardworks Ltd 2004 Sequences and rules Some sequences follow a simple rule that is easy to describe For example, this sequence 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, … continues by adding each time Each number in this sequence is one less than a multiple of three Other sequences are completely random For example, the sequence of winning raffle tickets in a prize draw In maths we are mainly concerned with sequences of numbers that follow a rule of 27 © Boardworks Ltd 2004 Naming sequences Here are the names of some sequences which you may know already: 2, 4, 6, 8, 10, Even Numbers (or multiples of 2) 1, 3, 5, 7, 9, Odd numbers 3, 6, 9, 12, 15, Multiples of 5, 10, 15, 20, 25 Multiples of 1, 4, 9, 16, 25, Square numbers 1, 3, 6, 10,15, Triangular numbers of 27 © Boardworks Ltd 2004 Ascending sequences When each term in a sequence is bigger than the one before the sequence is called an ascending sequence For example, The terms in this ascending sequence increase in equal steps by adding each time 2, 7, 12, 17, 22, 27, 32, 37, +5 +5 +5 +5 +5 +5 +5 The terms in this ascending sequence increase in unequal steps by starting at 0.1 and doubling each time 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, ×2 of 27 ×2 ×2 ×2 ×2 ×2 ×2 © Boardworks Ltd 2004 Descending sequences When each term in a sequence is smaller than the one before the sequence is called a descending sequence For example, The terms in this descending sequence decrease in equal steps by starting at 24 and subtracting each time 24, 17, 10, 3, –4, –11, –18, –25, –7 –7 –7 –7 –7 –7 –7 The terms in this descending sequence decrease in unequal steps by starting at 100 and subtracting 1, 2, 3, … 100, 99, 97, 94, 90, 85, 79, 72, –1 of 27 –2 –3 –4 –5 –6 –7 © Boardworks Ltd 2004 Sequences from real-life Number sequences are all around us Some sequences, like the ones we have looked at today follow a simple rule Some sequences follow more complex rules, for example, the time the sun sets each day Some sequences are completely random, like the sequence of numbers drawn in the lottery What other number sequences can be made from real-life situations? of 27 © Boardworks Ltd 2004 Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts 10 of 27 © Boardworks Ltd 2004 Finding the nth term of a linear sequence The terms in this sequence 5, 3, –2 1, –2 –1, –2 –3, –2 –5, –2 –7, –2 –9 … –2 can be found by subtracting each time Compare the terms in the sequence to the multiples of –2 Position × –2 Multiples of –2 –2 Term × –2 –4 +7 × –2 –6 +7 +7 × –2 –8 +7 –1 … × –2 × –2 –10 –2n +7 –3 n +7 … – 2n Each term is seven more than a multiple of –2 55 of 27 © Boardworks Ltd 2004 Arithmetic sequences Sequences that increase (or decrease) in equal steps are called linear or arithmetic sequences The difference between any two consecutive terms in an arithmetic sequence is a constant number When we describe arithmetic sequences we call the difference between consecutive terms, d We call the first term in an arithmetic sequence, a For example, if an arithmetic sequence has a = and d = -2, We have the sequence: 5, 56 of 27 3, 1, -1, -3, -5, © Boardworks Ltd 2004 The nth term of an arithmetic sequence The rule for the nth term of any arithmetic sequence is of the form: T(n) = an + b a and b can be any number, including fractions and negative numbers For example, T(n) = 2n + Generates odd numbers starting at T(n) = 2n + Generates even numbers starting at T(n) = 2n – Generates even numbers starting at –2 T(n) = 3n + Generates multiples of starting at T(n) = – n Generates descending integers starting at 57 of 27 © Boardworks Ltd 2004 Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term A4.5 Sequences from practical contexts 58 of 27 © Boardworks Ltd 2004 Sequences from practical contexts The following sequence of patterns is made from L-shaped tiles: Number of Tiles 12 16 The number of tiles in each pattern form a sequence How many tiles will be needed for the next pattern? We add on four tiles each time This is a term-to-term rule 59 of 27 © Boardworks Ltd 2004 Sequences from practical contexts A possible justification of this rule is that each shape has four ‘arms’ each increasing by one tile in the next arrangement The pattern give us multiples of 4: lot of lots of lots of 4 lots of The nth term is × n or 4n Justification: This follows because the 10th term would be 10 lots of 60 of 27 © Boardworks Ltd 2004 Sequences from practical contexts Now, look at this pattern of blocks: Number of Blocks 10 13 How many blocks will there be in the next shape? We add on blocks each time This is the term-to term rule Justification: The shapes have three ‘arms’ each increasing by one block each time 61 of 27 © Boardworks Ltd 2004 Sequences from practical contexts How many blocks will there be in the 100th arrangement? We need a rule for the nth term Look at pattern again: 1st pattern 2nd pattern 3rd pattern 4th pattern The nth pattern has 3n + blocks in it Justification: The patterns have ‘arms’ each increasing by one block each time So the nth pattern has 3n blocks in the arms, plus one more in the centre 62 of 27 © Boardworks Ltd 2004 Sequences from practical contexts So, how many blocks will there be in the 100 th pattern? Number of blocks in the nth pattern = 3n + When n is 100, Number of blocks = (3 × 100) + = 301 How many blocks will there be in: a) Pattern 10? (3 × 10) + = 31 b) Pattern 25? (3 × 25) + = 76 c) Pattern 52? (3 × 52) + = 156 63 of 27 © Boardworks Ltd 2004 Paving slabs 64 of 27 © Boardworks Ltd 2004 Paving slabs 65 of 27 © Boardworks Ltd 2004 Paving slabs The number of blue tiles form the sequence 8, 13, 18, 32, Pattern number Number of blue tiles 13 18 The rule for the nth term of this sequence is T(n) = 5n + Justification: Each time we add another yellow tile we add blue tiles The +3 comes from the tiles at the start of each pattern 66 of 27 © Boardworks Ltd 2004 Dotty pattern 67 of 27 © Boardworks Ltd 2004 Dotty pattern 68 of 27 © Boardworks Ltd 2004 Leapfrog investigation 69 of 27 © Boardworks Ltd 2004