Sequences

KS3 Mathematics : A4 Sequences KS3 Mathematics

Contents : A A A A A A4.1 Introducing sequences Contents A4 Sequences A4.5 Sequences from practical contexts A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term

Introducing sequences : In maths, we call a list of numbers in order a sequence. Each number in a sequence is called a 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. Introducing sequences

Infinite and finite sequences : A sequence can be infinite. That means it continues forever. The sequence of multiples of 10, Infinite and finite sequences 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. The sequence of two-digit square numbers 16, 25 ,36, 49, 64, 81 is finite.

Sequences and rules : Some sequences follow a simple rule that is easy to describe. For example, this sequence Sequences and rules 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, … continues by adding 3 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.

Naming sequences : Here are the names of some sequences which you may know already: 2, 4, 6, 8, 10, ... 1, 3, 5, 7, 9, ... 3, 6, 9, 12, 15, ... 5, 10, 15, 20, 25, ... 1, 4, 9, 16, 25, ... Even numbers (or multiples of 2) Odd numbers Multiples of 3 Multiples of 5 Square numbers 1, 3, 6, 10, 15, ... Triangular numbers Naming sequences

Ascending sequences : Ascending sequences When each term in a sequence is bigger than the one before the sequence is called an ascending sequence. The terms in this ascending sequence increase in equal steps by adding 5 each time. 2, 7, 12, 17, 22, 27, 32, 37, ... 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, ...

Descending sequences : Descending sequences When each term in a sequence is smaller than the one before the sequence is called a descending sequence. The terms in this descending sequence decrease in equal steps by starting at 24 and subtracting 7 each time. 24, 17, 10, 3, –4, –11, –18, –25, ... 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, ...

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. What other number sequences can be made from real-life situations? Sequences from real-life

Contents : A A A A A A4.2 Describing and continuing sequences Contents A4 Sequences A4.5 Sequences from practical contexts A4.1 Introducing sequences A4.3 Generating sequences A4.4 Finding the nth term

Sequences from geometrical patterns : We can show many well-known sequences using geometrical patterns of counters. Even Numbers Odd Numbers Sequences from geometrical patterns

Sequences from geometrical patterns : Multiples of Three Multiples of Five Sequences from geometrical patterns

Sequences from geometrical patterns : Square Numbers Triangular Numbers Sequences from geometrical patterns

Sequences with geometrical patterns : How could we arrange counters to represent the sequence 2, 6, 12, 20, 30, ...? The numbers in this sequence can be written as: 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, ... We can show this sequence using a sequence of rectangles: Sequences with geometrical patterns

Powers of two : Powers of two We can show powers of two like this: Each term in this sequence is double the term before it.

Powers of three : Powers of three We can show powers of three like this: Each term in this sequence is three times the term before it.

Sequences that increase in equal steps : Sequences that increase in equal steps We can describe sequences by finding a rule that tells us how the sequence continues. To work out a rule it is often helpful to find the difference between consecutive terms. For example, look at the difference between each term in this sequence: 3, 7, 11, 15, 19, 23, 27, 31, ... This sequence starts with 3 and increases by 4 each time. Every term in this sequence is one less than a multiple of 4.

Sequences that decrease in equal steps : Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 22 and decreases by 6 each time. Sequences that decrease in equal steps 22, 16, 10, 4, –2, –8, –14, –20, ... Each term in the sequence is two less than a multiple of 6. Sequences that increase or decrease in equal steps are called linear or arithmetic sequences.

Sequences that increase in increasing steps : Some sequences increase or decrease in unequal steps. Sequences that increase in increasing steps Look at the differences between terms in this sequence: 5, 6, 8, 11, 15, 20, 26, 33, ... This sequence starts with 5 and increases by 1, 2, 3, 4, … The differences between the terms form a linear sequence.

Sequences that decrease in decreasing steps : Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 7 and decreases by 0.1, 0.2, 0.3, 0.4, 0.5, … Sequences that decrease in decreasing steps 7, 6.9, 6.7, 6.4, 6, 5.5, 4.9, 4.2, ... With sequences of this type it is often helpful to find a second row of differences.

Using a second row of differences : Can you work out the next three terms in this sequence? Look at the differences between terms. Using a second row of differences 1, 3, 8, 16, 27, 41, 58, 78, ... A sequence is formed by the differences so we look at the second row of differences. This shows that the differences increase by 3 each time.

Sequences that increase by multiplying : Some sequences increase or decrease by multiplying or dividing each term by a constant factor. Sequences that increase by multiplying For example, look at this sequence: 2, 4, 8, 16, 32, 64, 128, 256, ... This sequence starts with 2 and increases by multiplying the previous term by 2. All of the terms in this sequence are powers of 2.

Sequences that decrease by dividing : Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 1024 and decreases by dividing by 4 each time. Sequences that decrease by dividing 1024, 256, 64, 16, 4, 1, 0.25, 0.0625, ...

Fibonacci-type sequences : Can you work out the next three terms in this sequence? How did you work these out? This sequence starts with 1, 1 and each term is found by adding together the two previous terms. Fibonacci-type sequences 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it.

Describing and continuing sequences : Describing and continuing sequences Here are some of the types of sequence you may come across: Sequences that increase or decrease by adding together the two previous terms. Sequences that increase or decrease in equal steps. These are called linear or arithmetic sequences. Sequences that increase or decrease in unequal steps by multiplying or dividing by a constant factor. Sequences that increase or decrease in unequal steps by adding or subtracting increasing or decreasing numbers.

Continuing sequences : A number sequence starts as follows 1, 2, ... Continuing sequences How many ways can you think of continuing the sequence? Give the next three terms and the rule for each one.

Finding missing terms : Finding missing terms

Name that sequence! : Name that sequence!

Contents : A A A A A A4.3 Generating sequences Contents A4 Sequences A4.5 Sequences from practical contexts A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.4 Finding the nth term

Sequence grid : Sequence grid

Generating sequences from flow charts : Generating sequences from flow charts A sequence can be given by a flow chart. For example: START Write down 3. Add on 1.5. Write down the answer. Is the answer more than 10? STOP This flow chart generates the sequence: 3, 4.5, 6, 6.5, 9, 10.5. This sequence has only six terms. It is finite.

Generating sequences from flow charts : START Write down 5. Subtract 2.1. Write down the answer. Is the answer less than –3? STOP Generating sequences from flow charts This flow chart generates the sequence 5, 2.9, 0.8, –1.3, –3.4.

Generating sequences from flow charts : START Write down 200. Divide by 2. Write down the answer. Is the answer less than 7? STOP Generating sequences from flow charts This flow chart generates the sequence 200, 100, 50, 25, 12.5, 6.25.

Generating sequences from flow charts : START Write down 3 and 4. Add together the two previous numbers. Write down the answer. Is the answer more than 100? STOP Generating sequences from flow charts This flow chart generates the sequence 3, 4, 7, 11, 18, 29, 47, 76, 123.

Predicting terms in a sequence : Predicting terms in a sequence Usually, we can predict how a sequence will continue by looking for patterns. We can predict that this sequence continues by subtracting 3 each time. However, sequences do not always continue as we would expect. For example: A sequence starts with the numbers 1, 2, 4, ... How could this sequence continue?

Continuing sequences : Here are some different ways in which the sequence might continue: 1 2 4 7 11 16 22 1 2 4 8 16 32 64 We can never be certain how a sequence will continue unless we are given a rule or we can justify a rule from a practical context. Continuing sequences

Continuing sequences : This sequence continues by adding 3 each time. We can say that rule for getting from one term to the next term is add 3. This is called the term-to-term rule. The term-to-term rule for this sequence is +3. Continuing sequences 1 4 7 10 13 16 19

Using a term-to-term rule : Does the rule +3 always produce the same sequence? No, it depends on the starting number. If we start with 2 and add on 3 each time we have, 2, 17, 20, 23, ... 5, 8, 11, 14, If we start with 0.4 and add on 3 each time we have, 0.4, 15.4, 18.4, 21.4, ... 3.4, 6.4, 9.4, 12.4, Using a term-to-term rule

Writing sequences from term-to-term-rules : Writing sequences from term-to-term-rules A term-to-term rule gives a rule for finding each term of a sequence from the previous term or terms. To generate a sequence from a term-to-term rule we must also be given the first number in the sequence. For example: 1st term 5 Term-to-term rule Add consecutive even numbers starting with 2. This gives us the sequence, 5 7 11 17 25 35 47 ...

Sequences from a term-to-term rule : Write the first five terms of each sequence given the first term and the term-to-term rule. 1st term Term-to-term rule 21 80 48 50000 –1 1.2 Sequences from a term-to-term rule 10 100 3 5 7 0.8 Add 3 Subtract 5 Double Multiply by 10 Subtract 2 Add 0.1 10, 13, 16, 19, 100, 95, 90, 85, 3, 6, 12, 24, 5, 50, 500, 5000, 7, 5, 3, 1, 0.8, 0.9, 1.0, 1.1,

Sequences from position-to-term rules : Sometimes sequences are arranged in a table like this: Sequences from position-to-term rules We can say that each term can be found by multiplying the position of the term by 3. This is called a position-to-term rule. For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence. What is the 100th term in this sequence? 3 × 100 = 300 3n … 18 15 12 9 6 3

Sequences from position-to-term rules : Sequences from position-to-term rules

Writing sequences from position-to-term rules : Writing sequences from position-to-term rules The position-to-term rule for a sequence is very useful because it allows us to work out any term in the sequence without having to work out any other terms. We can use algebraic shorthand to do this. We call the first term T(1), for Term number 1, we call the second term T(2), we call the third term T(3), ... and we call the nth term T(n). T(n) is called the the nth term or the general term.

Writing sequences from position-to-term rules : For example, suppose the nth term of a sequence is 4n + 1. We can write this rule as: T(n) = 4n + 1 Find the first 5 terms. T(1) = 4 × 1 + 1 = 5 T(2) = 4 × 2 + 1 = 9 T(3) = 4 × 3 + 1 = 13 T(4) = 4 × 4 + 1 = 17 T(5) = 4 × 5 + 1 = 21 The first 5 terms in the sequence are: 5, 9, 13, 17 and 21. Writing sequences from position-to-term rules

Writing sequences from position-to-term rules : If the nth term of a sequence is 2n2 + 3. We can write this rule as: T(n) = 2n2 + 3 Find the first 4 terms. T(1) = 2 × 12 + 3 = 5 T(2) = 2 × 22 + 3 = 11 T(3) = 2 × 32 + 3 = 21 T(4) = 2 × 42 + 3 = 35 The first 4 terms in the sequence are: 5, 11, 21, and 35. Writing sequences from position-to-term rules This sequence is a quadratic sequence.

Sequence generator – linear sequences : Sequence generator – linear sequences

Sequence generator – non-linear sequences : Sequence generator – non-linear sequences

Sequences and rules : Which rule is best? The term-to-term rule? The position-to-term rule? Sequences and rules

Contents : A A A A A A4.4 Finding the nth term Contents A4 Sequences A4.5 Sequences from practical contexts A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences

Sequences of multiples : Sequences of multiples All sequences of multiples can be generated by adding the same amount each time. They are linear sequences. For example, the sequence of multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, … can be found by adding 5 each time. Compare the terms in the sequence of multiples of 5 to their position in the sequence: Position Term 1 5 2 10 3 15 4 20 5 25 n … … 5n

Sequences of multiples : Sequences of multiples The sequence of multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, … can be found by adding 3 each time. Compare the terms in the sequence of multiples of 3 to their position in the sequence: Position Term 1 3 2 6 3 9 4 12 5 15 n … … 3n The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.

Sequences of multiples : Sequences of multiples For example: The nth term of 4, 8, 12, 16, 20, 24, … is 4n The 10th term of this sequence is 4 × 10 = 40 The 25th term of this sequence is 4 × 25 = 100 The 47th term of this sequence is 4 × 47 = 188

Finding the nth term of a linear sequence : Finding the nth term of a linear sequence The terms in this sequence 4, 7, 10, 13, 16, 19, 22, 25, … can be found by adding 3 each time. Compare the terms in the sequence to the multiples of 3. Position Multiples of 3 1 2 3 4 5 n … 3n Term 4 7 10 13 16 … 3 6 9 12 15 3n + 1 Each term is one more than a multiple of 3.

Finding the nth term of a linear sequence : Finding the nth term of a linear sequence The terms in this sequence 1, 6, 11, 16, 21, 26, 31, 36, … can be found by adding 5 each time. Compare the terms in the sequence to the multiples of 5. Position Multiples of 5 1 2 3 4 5 n … 5n Term 1 6 11 16 21 … 5 10 15 20 25 5n – 4 Each term is four less than a multiple of 5.

Finding the nth term of a linear sequence : Finding the nth term of a linear sequence The terms in this sequence 5, 3, 1, –1, –3, –5, –7, –9, … can be found by subtracting 2 each time. Compare the terms in the sequence to the multiples of –2. Position Multiples of –2 1 2 3 4 5 n … –2n Term 5 3 1 –1 –3 … –2 –4 –6 –8 –10 7 – 2n Each term is seven more than a multiple of –2.

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. If an arithmetic sequence has a = 5 and d = –2, We have the sequence: 5, 3, 1, –1, –3, –5, ... Arithmetic sequences

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 + 1 Generates odd numbers starting at 3. T(n) = 2n + 4 Generates even numbers starting at 6. T(n) = 2n – 4 Generates even numbers starting at –2. T(n) = 3n + 6 Generates multiples of 3 starting at 9. T(n) = 4 – n Generates descending integers starting at 3. The nth term of an arithmetic sequence

Contents : A A A A A A4.5 Sequences from practical contexts Contents A4 Sequences A4.1 Introducing sequences A4.2 Describing and continuing sequences A4.3 Generating sequences A4.4 Finding the nth term

Sequences from practical contexts : The following sequence of patterns is made from L-shaped tiles: Number of Tiles 4 8 12 The number of tiles in each pattern forms a sequence. How many tiles will be needed for the next pattern? 16 We add on four tiles each time. This is a term-to-term rule. Sequences from practical contexts

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: 1 lot of 4 2 lots of 4 3 lots of 4 4 lots of 4 The nth term is 4 × n or 4n. Justification: This follows because the 10th term would be 10 lots of 4. Sequences from practical contexts

Sequences from practical contexts : Now, look at this pattern of blocks: Number of Blocks 4 7 10 How many blocks will there be in the next shape? 13 We add on 3 blocks each time. This is the term-to term rule. Justification: The shapes have three ‘arms’ each increasing by one block each time. Sequences from practical contexts

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 + 1 blocks in it. Justification: The patterns have 3 ‘arms’ each increasing by one block each time. So the nth pattern has 3n blocks in the arms, plus one more in the centre. Sequences from practical contexts

Sequences from practical contexts : So, how many blocks will there be in the 100th pattern? Number of blocks in the nth pattern = 3n + 1 When n is 100, Number of blocks = (3 × 100) + 1 = How many blocks will there be in the: a) 10th pattern? (3 × 10) + 1 = b) 25th pattern? (3 × 25) + 1 = c) 52th pattern? (3 × 52) + 1 = 301 31 76 157 Sequences from practical contexts

Paving slabs 1 : Paving slabs 1

Paving slabs 2 : The number of blue tiles in a pattern form the sequence 8, 13, 18, 23, ... Pattern number 1 Number of blue tiles 8 2 13 3 18 The rule for the nth term of this sequence is T(n) = 5n + 3 Justification: Each time we add another yellow tile we add 5 blue tiles. The +3 comes from the 3 tiles at the start of each pattern. Paving slabs 2

Dotty pattern 1 : Dotty pattern 1

Leapfrog investigation : Leapfrog investigation