MEP Pupil Text 12. A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued.


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1 MEP Pupil Text Number Patterns. Simple Number Patterns A list of numbers which form a pattern is called a sequence. In this section, straightforward sequences are continued. Worked Example Write down the next three numbers in each sequence., 4, 6, 8, 0,... 3, 6, 9,, 5,... This sequence is a list of even numbers, so the next three numbers will be, 4, 6. This sequence is made up of the multiples of 3, so the next three numbers will be 8,, 4. Worked Example Find the next two numbers in each sequence. 6, 0, 4, 8,,... 3, 8, 3, 8, 3,... For this sequence the difference between each term and the next term is 4. Sequence 6, 0, 4, 8,,... Difference So 4 must be added to obtain the next term in the sequence. The next two terms are + 4 = 6 and = 30, giving 6, 0, 4, 8,, 6, 30,... For this sequence, the difference between each term and the next is 5. Sequence 3, 8, 3, 8, 3,... Difference Adding 5 gives the next two terms as = 8 and = 33, giving 3, 8, 3, 8, 3, 8, 33,... 6
2 MEP Pupil Text Exercises. Write down the next four numbers in each list., 3, 5, 7, 9,... 4, 8,, 6, 0,... (c) 5, 0, 5, 0, 5,... (d) 7, 4,, 8, 35,... (e) 9, 8, 7, 36, 45,... (f) 6,, 8, 4, 30,... (g) 0, 0, 30, 40, 50,... (h),, 33, 44, 55,... (i) 8, 6, 4, 3, 40,... (j) 0, 40, 60, 80, 00,... (k) 5, 30, 45, 60, 75,... (l) 50, 00, 50, 00, 50,.... Find the difference between terms for each sequence and hence write down the next two terms of the sequence. 5, 8,, 4, 7,..., 0, 8, 6, 34,... (c) 7,, 7,, 7,... (d) 6, 7, 8, 39, 50,... (e) 8, 5,, 9, 36,... (f), 3, 3, 4, 4,... (g) 4, 3,, 3, 40,... (h) 6, 3, 0, 7, 4,... (i) 0, 6,, 8, 4,... (j) 8, 4, 0, 6,,... (k), 8, 5,,,... (l) 5, 8,, 4, 7, In each part, find the answers to (i) to (iv) with a calculator and the answer to (v) without a calculator. (i) =? (i) 99 =? (ii) =? (ii) 999 =? (iii) =? (iii) 9999 =? (iv) =? (iv) =? (v) =? (v) =? (c) (i) 88 =? (d) (i) 7 9 =? (ii) 888 =? (ii) 7 99 =? (iii) 8888 =? (iii) =? (iv) =? (iv) =? (v) =? (v) =? 4. (i) Complete the following number pattern: = = =?? =? (ii) Look at the numbers in the right hand column. Write down what you notice about these numbers. 63
3 . MEP Pupil Text Use your calculator to work out the next line of the pattern. What do you notice now? (NEAB) 5. 9 = = = = = = = = 8 (i) Complete the statement, The units digits in the right hand column, in order, are 8, 7, 6,... (ii) (iii) Complete the statement, The tens digits in the righthand column, in order, are,, 3,... What is the connection between the answers to parts (i) and (ii)? The numbers in the righthand column go up by 9 each time. What else do you notice about these numbers? (MEG). Recognising Number Patterns This section looks at how the terms of a sequence are related. For example the Fibonacci sequence:,,, 3, 5, 8, 3,... is obtained by adding together two consecutive terms to obtain the next term. Worked Example + = + = = = = 3 The sequence of square numbers is, 4, 9, 6, 5, 36,... Explain how to obtain the next number in the sequence. The next number can be obtained in one of two ways. 64
4 MEP Pupil Text () The sequence could be written,,, 3, 4, 5, 6,... and so the next term would be 7 = 49. () Calculating the differences between the terms gives Sequence, 4, 9, 6, 5, 36,... Difference The difference increases by each time so the next term would be = 49. Worked Example Describe how to obtain the next term of each sequence below. 3, 0, 7, 4, 3,... 3, 6,, 8, 7,... (c), 5, 6,, 7, 8,... Finding the differences between the terms gives Sequence 3, 0, 7, 4, 3,... Difference All the differences are the same so each term can be obtained by adding 7 to the previous term. The next term would be = 38. Here again the differences show a pattern. Sequence 3, 6,, 8, 7,... Difference Here the differences increase by each time, so to find further terms add more than the previous difference. The next term would be 7 + = 38. (c) The differences again help to see the pattern for this sequence. Sequence, 5, 6,, 7, 8,... Difference If the first difference, 4, is ignored, the remaining pattern of the differences is the same as the sequence itself. This shows that the difference between any two terms is equal to the previous term. So a new term is obtained by adding together the two previous terms. The next term of the sequence will be =
5 . MEP Pupil Text Worked Example 3 A sequence of shapes is shown below. Write down a sequence for the number of line segments and explain how to find the next number in the sequence. The sequence for the number of line segments is 4, 8,, 6,... The difference between each pair of terms is 4, so to the previous term add 4. Then the next term is = 0. This corresponds to the shape opposite, which has 0 line segments. Exercises. Find the next two terms of each sequence below, showing the calculations which have to be done to obtain them. 5,, 7, 3, 9,... 6, 0, 5,, 8,... (c), 9, 6, 3, 0,... (d) 30,, 5, 9, 4,... (e) 50, 56, 63, 7, 80,... (f),, 4, 8, 4,,... (g) 6,, 6,, 6,... (h) 0, 3, 8, 5, 4,... (i) 3, 6,, 4, 48,... (j), 4, 0,, 46,... (k) 4,,, 6, 46,.... A sequence of numbers is, 8, 7, 64, 5,... By considering the differences between the terms, find the next two terms. 3. Each sequence of shapes below is made up of lines which join two points. For each sequence: (i) (ii) (iii) write down the number of lines, as a sequence; explain how to obtain the next term of the sequence; draw the next shape and check your answer. 66
6 MEP Pupil Text (c) (d) (e) 4. Write down a sequence for the number of dots in each pattern. Then explain how to get the next number. (c) (d) 67
7 . MEP Pupil Text (e) (f) 5. Describe how the sequence is formed., 4, 9, 6, 5, What is the relationship between the sequence in and the sequences below? For each sequence explain how to calculate the terms. (i) 3, 6,, 8, 7, 38,... (ii),, 7, 4, 3, 34,... (iii), 8, 8, 3, 50, 7,... (iv) 4, 6, 36, 64, 00, 44, Describe how the sequences below are related. (i),,, 3, 5, 8, 3,,... (ii) 4, 4, 5, 6, 8,, 6, 4,... (iii),, 0,, 3, 6,, 9,... Describe how to find the next term of each sequence. 7. A computer program prints out the following numbers When one of these numbers is changed, the numbers will form a pattern. Circle the number which has to be changed and correct it. Give a reason why your numbers now form a pattern. (SEG) 8. Here are the first four numbers of a number pattern. 7, 4,, 8,... Write down the next two numbers in the pattern. Describe, in words, the rule for finding the next number in the pattern. (LON) 68
8 MEP Pupil Text 9. To generate a sequence of numbers, Paul multiplies the previous number in the sequence by 3, then subtracts. Here are the first four numbers of his sequence.,, 5, 4,... Find the next two numbers in the sequence. Here are the first four numbers of the sequence of cube numbers., 8, 7, 64,... Find the next two numbers in the sequence. (MEG) 0. Row q Row p The numbers in Row of the above pattern are found by using pairs of numbers from Row. For example, 7 = 9 9 = 8 9 By considering the sequence in Row, write down the value of p. Find the value of q. (MEG). A number pattern begins 4, 8,, 6, 0, 4,... Describe the number pattern. Another number pattern begins, 4, 9, 6, 5, 36,... (i) Describe this number pattern. (ii) What is the next number in this pattern? Each number in this pattern is changed to make a new number pattern. The new number pattern begins (iii),, 7, 4, 3, 34,... What is the next number in the new pattern? Explain how you found your answer. (SEG). (i) Write down the multiples of 5, from 5 to 40. (ii) Describe the pattern of the units digits. Sequence P is 3, 6, 9,, 5, 8,,... Explain how sequence P is produced. 69
9 . MEP Pupil Text (c) Copy and complete the table below. Sequence P Add and then multiply by Sequence Q = 4, 4 = = 7, 7 = 4 4 9?????? 5??? 8??? (d) (i) Find the next two terms in the sequence, 4, 0, 9, 3, 46, 64,... (ii) Explain how you obtained your answer to part (d) (i). (MEG).3 Extending Number Patterns A formula or rule for extending a sequence can be used to work out any term of a sequence without working out all the terms. For example, the 00th term of the sequence,, 4, 7, 0, 3,... can be calculated as 98 without working out any other terms. Worked Example Find the 0th term of the sequence 8, 6, 4, 3,... The terms of the sequence can be obtained as shown below. This pattern can be extended to give Worked Example st term = 8 = 8 nd term = 8 = 6 3rd term = 3 8 = 4 4th term = 4 8 = 3 0th term = 0 8 = 60 Find the 0th and 00th terms of the sequence 3, 5, 7, 9,,... 70
10 MEP Pupil Text The terms above are given by st term = 3 nd term = 3 + = 5 3rd term = 3 + = 7 4th term = = 9 This can be extended to give 5th term = = 0th term = = 00th term = = 0. Worked Example 3 Find the 0th term of the sequence, 5, 0, 7, 6, 37,... The terms of this sequence can be expressed as st term = + nd term = + 3rd term = 3 + 4th term = 4 + Extending the pattern gives 5th term = 5 + 0th term = 0 + = 40. Exercises. Find the 0th and 0th terms of each sequence below. 4, 8,, 6, 0,... 5, 0, 5, 0, 5,... (c),, 3, 4, 5,... (d) 7, 9,, 3, 5,... (e) 5, 9, 3, 7,,... (f) 0, 9, 8, 7, 6,... (g) 50, 44, 38, 3, 6,... (h), 5, 8, 3,... (i) 8, 7, 6, 5, 4,... (j) 4, 0, 4, 8,,... (k) 7,, 7,, 7,... (l) 3,, 7,,... 7
11 .3 MEP Pupil Text. Find the 0th term for each of the two sequences below. (i) 3, 6,, 8, 7,... (ii) 5, 6, 7, 8, 9,... Hence find the 0th term of the sequences, (i) 8,, 8, 6, 36,... (ii) 5, 36, 77, 44, 43,... (iii) 0, 4, 0, 8, Find the 0th term of the sequences below. (i) 4, 9, 4, 9, 4,... (ii) 3, 5, 7, 9,,... Use your answers to to find the 0th term of the sequence, 45, 98, 7, 64, By considering the two sequences, 4, 9, 6, 5,...,, 3, 4, 5,... find the 0th term of the sequence 0,, 6,, 0,... Find the 0th term of the sequence 0, 6, 4, 60, 0, For each sequence of shapes below find the number of dots in the 0th shape. (c) 6. Patterns of triangles are made using sticks. The first three patterns are drawn below. Pattern Pattern Pattern 3 7
12 MEP Pupil Text Pattern number 3 Number of sticks How many sticks has Pattern 4? A pattern needs 33 sticks. What is the number of this pattern? (c) (i) How many sticks are needed to make Pattern 00? (ii) Explain how you found your answer. (SEG).4 Formulae and Number Patterns This section considers how the terms of a sequence can be found using a formula and how a formula can be found for some simple sequences. The terms of a sequence can be described as u, u, u 3, u 4, u 5,... where u is the first term, u is the second term and so on. Consider the sequence, 4, 9, 6, 5,... u = = u = 4 = u 3 = 9 = 3 3 u 4 = 6 = 4 4 u 5 = 5 = 5 5 This sequence can be described by the general formula Worked Example un = n. Find the first 5 terms of the sequence defined by the formula un = 3n + 6. u = = 9 u = = u 3 = = 5 u 4 = = 8 u 5 = = So the sequence is 9,, 5, 8,,... Note that the terms of the sequence increase by 3 each time and that the formula contains a ' 3n '. 73
13 .4 MEP Pupil Text Worked Example Find the first 5 terms of the sequence defined by the formula un = 5n 4. u = 5 4 = u = 5 4 = 6 u 3 = = u 4 = = 6 u 5 = = So the sequence is, 6,, 6,,... Here the terms increase by 5 each time and the formula contains a ' 5n '. In general, if the terms of a sequence increase by a constant amount, d, each time, then the sequence will be defined by the formula where c is a constant number. Worked Example 3 un = dn + c Find a formula to describe each of the sequences below. 3, 0, 7, 34, 4, 48,...,, 3, 34, 45, 56,... First find the differences between the terms. Sequence 3, 0, 7, 34, 4, Difference As the difference between each term is always 7, the formula will contain a ' 7n' and be of the form un = 7 n + c. To find the value of c consider any term. The first one is usually easiest to use. Here, using n = and u = 3 gives so the formula is 3 = 7 + c 3 = 7 + c c = 6, un = 7 n
14 MEP Pupil Text Note Always check that the formula is correct for other terms, e.g. n = and n = 3. In this case, u = = 0 so the formula holds. and u 3 = = 7 Again start by finding the differences between the terms of the sequence. Sequence,, 3, 34, 45, 56,... Difference The difference is always so the formula will contain 'n' and will be un = n + c. Using the first term, i.e. n = and u =, gives so the formula is = + c = + c c = 0, un = n 0. [Check: u = 0 = and u 3 = 3 0 = 3, which are correct.] Worked Example 4 The 7th, 8th and 9th terms of a sequence are 6, 69 and 77 respectively. Find a formula to describe this sequence. Looking at the differences, Sequence 6, 69, 77, Difference 8 8 you can conclude that the sequence must be of the form un = 8 n + c. For n = 7, u7 = 56 + c, giving c = 5. Thus the formula to describe the sequence is un = 8n
15 .4 MEP Pupil Text Note A more general way of tackling this type of problem is to fit the linear sequence to the given information. un = dn + c In this case, u7 = 7d + c = 6 and u8 = 8d + c = 69. Subtracting u 7 from u 8 gives d = 8 and substituting for d in either of the two equations gives c = 5. Thus, as before, un = 8n + 5. Exercises. Use the formulae below to find the first 6 terms of each sequence. un = 4n + un = 5n 7 (c) un = 0n + (d) un = n (e) un = n + n (f) u n =. The sequences described by the formulae, un = 8n, un = 3n+ 5, un = n +, un = 3 n are given below. Select the formula that describes each sequence. 0, 7, 6, 63, 4,... 8,, 4, 7, 0,... (c) 6, 4,, 30, 38,... (d), 5, 0, 7, 6, Find the 0th term for each sequence below. un = 4 n u n = 3 n 50 (c) un = 4 n + 7 (d) u n 40 (e) u n n = n = n 4 (f) u n = Consider the formula for the sequence below. 8, 5,, 9, 36, Explain why the formula contains 7n. Find the formula for the sequence. 76
16 MEP Pupil Text 5. Find the formula which describes each sequence below. 4, 9, 4, 9, 4,..., 4, 7, 0, 3,... (c), 4, 0, 6,,... (d) 00, 98, 96, 94, 9,... (e),,,, 3,... (f) 5,, 9, 6, 3,... (g) 0, 9, 9, 8, 8, Which of the sequences below are described by a formula of the form un = dn + c? Where possible, give the formula.,,, 3, 5, 8,..., 0, 8, 8, 46, 74,... (c) 0, 3, 8, 5, 4, 35, 48,... (d),, 4, 7,, 6,... (e),.,.,.3,.4,.5,... (f),.,.,.33,.464,.605,... (g) 3, 7,, 5, 9, 3, Write down the first 6 terms of the sequence un n. Then use your answer to write down formulae for the following sequences. 3, 6,, 8, 7, 38,... 4,, 4,, 0, 3,... (c), 8, 8, 3, 50, 7,... (d) 0, 6, 6, 30, 48, 70,... (e) 99, 96, 9, 84, 75, 64, By considering the sequence described by un n, find formulae to describe the following sequences. 0, 7, 6, 63, 4,... 0, 7, 36, 73, 34,... (c) 99, 9, 73, 36, 75, The 0th, th and th terms of a sequence are 50, 54 and 58. Find a formula to describe this sequence and write down the first 5 terms. 0. The 00th, 0st and 0nd terms of a sequence are 608, 64 and 60. Find a formula to describe this sequence and find the 0th term.. The 0th, th, 4th and 6th terms of a sequence are 5, 6, 7 and 8. Find a formula to describe this sequence and find its first term. 77
17 .4 MEP Pupil Text. Consider the sequence,, 5, 9, 3, 7,, 5,... Find the next term in the sequence and explain how you obtained your answer. The n th term in the sequence is 4n 3. Solve the equation 4 n 3 = 397 and explain what the answer tells you. (MEG) 3. Here are the first four terms of a number sequence. 7,, 5, 9, Write down the n th term of the sequence. (LON) 4. Sheep enclosures are built using fences and posts. The enclosures are always built in a row. Post Fence One enclosure Two enclosures Three enclosures Sketch (i) four enclosures in a row (ii) five enclosures in a row. Copy and complete the table below. Number of enclosures Number of posts 6 9 (c) (d) Work out the number of posts needed for 0 enclosures in a row. Write down an expression to find the number of posts needed for n enclosures in a row. (LON) 5. Patterns of squares are formed using sticks. The first three patterns are drawn below. Pattern Pattern Pattern 3 The table shows the number of sticks needed for each pattern. 78
18 MEP Pupil Text Pattern 3 Number of sticks (i) Draw Pattern 4. (ii) How many sticks are needed for Pattern 4? How many more sticks are needed to make Pattern 5 from Pattern 4? (c) There is a rule for finding the number of sticks needed to make any of these patterns of squares. If the number of squares in a pattern is s, write down the rule. (SEG) 6. Sticks are arranged in shapes. Shape Shape Shape 3 7 sticks sticks 7 sticks The number of sticks form a sequence. (i) Write down a rule for finding the next number in the sequence. (ii) Find a formula in terms of n for the number of sticks in the n th shape. Find a formula, in terms of n, for the area of the n th rectangle in this sequence (SEG) Just for Fun John and Julie had a date one Saturday. They agreed to meet outside the cinema at 8 pm. Julie thought that her watch was 5 minutes fast but in actual fact it was 5 minutes slow. John thought that his watch was 5 minutes slow but in actual fact it was 5 minutes fast. Julie deliberately turned up 0 minutes late while John decided to turn up 0 minutes early. Who turned up first and how long had he/she to wait for the other to arrive? 79
19 MEP Pupil Text.5 General Laws This section considers sequences which are formed in various ways and uses iterative formulae that describe how one term is obtained from the previous term. The behaviour of the sequences with huge numbers of terms is also considered to see whether they increase indefinitely or approach a fixed value. Worked Example Find a formula to generate the terms of these sequences: 3, 3 5, 4 7, 5 9, 6,... 4, 6, 9, 3.5, 0.5,... What happens to these sequences for large values of n? This can be approached by looking at the numerators and denominators of the fractions. The numbers, 3, 4, 5, 6,... are from the sequence un= n +. The numbers 3, 5, 7, 9,,... are from the sequence un = n +. Combining these gives the formula for the sequence as 3, 3 5, 4 7, 5 9, 6,... u n = n + n + As the value of n becomes larger and larger, this sequence produces terms that get closer and closer to. Consider the terms below: 0 u 00 = = u 000 = = u 0000 = = u = = We say that the sequence converges to. 80
20 MEP Pupil Text Note A more rigorous approach is to divide both the numerator and denominator of the expression for u n by n, giving n un = + = n + Now as n becomes larger, the term 0 n, giving u n = + n. + n We write u n as n (infinity) and say that ' u n tends to as n tends to infinity.' The terms of this sequence are multiplied by a factor of.5 to obtain the next term Considering each term helps to see the general formula. 0 u = 4 = 4 5. u = 4. 5 = 4. 5 u 3 = = u 4 = = u 5 = = 4 5. So the general term is u n n 4 5 =.. The terms of this sequence become larger and larger, never approaching a fixed value as in the last example. We say the sequence diverges. In fact, any sequence which does not converge is said to diverge. Worked Example Find iterative formulae for each of the following sequences. 7,, 5, 9, 3, 7,... 6,, 4, 48, 96,... (c),,, 3, 5, 8, 3,,... 8
21 .5 MEP Pupil Text This group of sequences show how one term is related to the previous term, so the formulae give u n+ in terms of the previous term, u n. u = 7 u = 6 u u u = = u + 4 u = 6 = u = + 4 = u + 4 u = = u = = u + 4 u = 4 = u So u (c) u = u = n = un + 4 with u = 7. So un+ = un with u = 6. u = + = u + u 3 u = + = u + u 4 3 u = + 3 = u + u u = = u + u So un+ = un + un, with u = and u =. Worked Example 3 Find the first 4 terms of the sequence defined iteratively by un+ = un + 4 u n starting with u =. Show that the sequence converges to. u = u 4 = + 5 =. u = = 05. u = = These terms appear to be getting closer and closer to. If the sequence does converge to a particular value then as n becomes larger and larger, u n+ becomes approximately the same as u n. So if un+ = un = d, say, then u n + = un + 4 u n 8
22 MEP Pupil Text becomes d = 4 d + d d d = d + = 4 d d = 4 4 d d = or. So the sequence does converge to. If the sequence had started with u =, then it would have converged to, the other possible value of d. Exercises. Find formulae to generate the terms of each sequence below. 0, 4, 9, 6 3, 5 4,... 5, 7 3, 9 4, 5, 3 6,... (c), 6, 8, 54, 6,... (d), 0.9, 0.8, 0.79, 0.656,... (e),.,.44,.78,.0736,... (f), 3, 9, 7, 8,... (g), 4, 8, 6, 3,... (h) 3, 5, 9, 7, 33,... (i) 3 4, 6 5, 9 6, 7, 5 8,... Which of the above sequences converge and which diverge? For those which converge, find the value to which they converge.. Find iterative formulae for each of the following sequences. 8, 5,,, 4,... 5, 0, 80, 30, 80,... (c), 3, 5, 9, 7,... (d) 4000, 000, 000, 500, 50,... (e) 3, 3, 6, 9, 5, 4, 39,... (f),,, 3, 5, 9, 7, 3, The iterative formula un+ = un + 6 u n can be used with u = to define the sequence. Find the first 5 terms of the sequence. Show that the sequence converges to 6. 83
23 .5 MEP Pupil Text 4. Does the sequence defined by un+ = un u with u = converge? If it does, find the value to which it converges. What happens if u is a different value? n 5. Find the first 5 terms of the sequence u + = u 7u ( ) n n n starting with u = 0.. To what value does this sequence converge? (Give your answer as a fraction.) 6. To what value does the sequence ( ) u + = u 3u n n n with u =, converge? Calculate the first four terms of the sequence to check your answer. 7. Javid and Anita try to find different ways of exploring the sequence 4, 0, 8, 8, 40,... Javid writes ( ) ( ) ( ) ( ) st number 4 = 4 = + 3 nd number 0 = 5 = + 3 3rd number 8 = 3 6 = th number 8 = 4 7 = How would Javid write down (i) the 5th number (ii) the nth number? Anita writes st number 4 = 3 nd number 0 = 3 4 3rd number 8 = 4 5 4th number 8 = 5 6 How would Anita write down (i) the 5th number (ii) the nth number? (c) Show how you would prove that Javid's expression and Anita's expression for the nth number are the same. (MEG) 8. Write down the next number in this sequence.,, 4, 8, 6, 3,... Describe how the sequence is formed. (c) One number in the sequence is 04. Describe how you can use the number 04 to find the number in the sequence which comes just before it. (SEG) 84
24 MEP Pupil Text 9. Row Sum = Row 3 5 Sum = 8 = 3 Row Sum = 7 = 3 3 Write down the numbers and sum which continue the pattern in Row 4. Which row will have a sum equal to 000? (c) What is the sum of Row 0? (d) The first number in a row is x. What is the second number in this row? Give your answer in terms of x. (MEG).6 Quadratic Formulae Consider the sequence generated by the formula un = n + n., 6,, 0, 30, 4,... The differences between terms can be considered as below., 6,, 0, 30, 4,... First differences Second differences The first differences increase, but the second differences are all the same. Whenever the second differences are constant a sequence can be described by a quadratic formula of the form un = an + bn + c where a, b and c are constants. Worked Example Find the formula which describes the sequence, 8,, 40, 65, 96. First examine the differences., 8,, 40, 65, 96,... First differences Second differences As the second differences are constant, the sequence can be described by a quadratic formula of the form un = an + bn + c. To find the values of a, b and c consider the first 3 terms. 85
25 .6 MEP Pupil Text Using u = gives = a + b + c () Using u = 8 gives 8 = 4a + b + c () Using u 3 = gives = 9a + 3b + c (3) These are three simultaneous equations. To solve them, subtract equation () from equation () to give equation (4) and from equation (3) to give equation (5) as below. 8 = 4a + b + c () = 9a + 3b + c (3) subtract = a + b + c () = a + b + c () 7 = 3a + b (4) 0 = 8a + b (5) Then subtracting equation (4) from equation (5) gives 0 = 8a + b (5) 4 = 6a + b (4) 6 = a so a = 3. Substituting for a in equation (4) gives 7 = b so b =. Finally, substituting for a and b in equation () gives = 3 + c so c = 0. The sequence is then generated by the formula Note un = 3n n. You should, of course, check the first few terms: n = u = 3 = n = u = 3 = 8 n = 3 u = =. 3 Investigation Find the next term in the sequence,, 9 4, 7 5, 7 86
26 MEP Pupil Text Alternative Approach Solving sets of simultaneous equations like these can be quite hard work. Examining carefully the differences leads to an easier method. The first four terms of the sequence are un = an + bn + c u = a + b + c u = 4a + b + c u3 = 9a + 3b + c u4 = 6a + 4b + c Consider the differences for these terms. a + b + c, 4a + b + c, 9a + 3b + c, 6a + 4b + c First difference 3a + b 5a + b 7a + b Second difference a a Note that the second difference is equal to a, the first of the first differences is 3a + b and the first term is a + b + c. This can be used to create a much easier approach to finding a, b and c, as shown in the next example. Worked Example Find the formula which generates the sequence 6,, 8, 7, 38, 5,... First find the differences. 6,, 8, 7, 38, 5,... First differences Second differences As the second differences are constant, the sequence is generated by the quadratic formula un = an + bn + c. Using the results from the differences considered in the alternative approach in Worked Example gives a = 3a + b = 5 a + b + c = 6. This gives a =, b =, and c = 3, so the formula is un = n + n
27 .6 MEP Pupil Text Exercises. Show that each sequence below has a constant second difference and use this to find the next terms. 5, 0, 7, 6, 37,... 0, 0, 8, 54, 88,... (c), 4, 34, 6, 98,... (d), 4, 6, 0, 4,... (e),,,, 7,.... The third, fourth and fifth terms of a quadratic sequence are 6, 0 and 6. Find the first, second and sixth terms of the sequence. 3. A sequence begins, 4, 4, 66,... Find the 0th term of this sequence. 4. The first and third terms of a sequence are 6 and 48. If the second difference is constant and equal to 0, find the second term of the sequence. 5. For each of the following sequences, state whether or not they are generated by a quadratic formula and if they are, give the formula. 3,, 3, 6,, 8,... 0, 4, 6, 66, 36, 600,... (c), 3, 7, 3,, 3,... (d), 3, 4, 7,, 8,... (e) 5, 0, 7, 6, 7, 40,... (f), 3, 7,, 5, 9,... (g),, 7, 9, 37, 6, The nd, 3rd and 4th terms of a quadratic sequence are 0, 3 and 8. Find the st and 5th terms of the sequence. 7. In a sequence, the nd term is 0 more than the st term, the 3rd term is 5 more than the nd term, the 4th term is 0 more than the 3rd term. Show that the sequence is quadratic and find a formula for the sequence if the first term is. 8. The terms of a particular cubic sequence are given by u n. Find the first 6 terms of this sequence and then the first, second and third differences. What do you notice? n = 3 Check that the result you noted in Part is true for a cubic sequence of your own choice. 3 (c) By considering un = an + bn + cn + d, show that the third difference of any cubic sequence is a constant and give its value in terms of a. 88
28 MEP Pupil Text (d) Then find the formula which describes the cubic sequence 4, 5, 40, 85, 56, 59, The table shows how the first 5 triangle, square and pentagon numbers are formed Triangle Square Pentagon 5 35 Show that all the sequences formed are quadratic and find expressions for them. The hexagon numbers give the sequence, 6, 5, 8, 45,... Show that the terms of this sequence are given by un = n n. (c) By looking at the expressions you have obtained so far, predict formulae for the heptagonal and octagonal numbers. Use the facts that the 8th heptagonal number is 48 and the 8th octagonal number is 76 to check your formulae. (d) Show that the 8th decagonal number is Look at the three sequences below. Sequence p 4, 6, 8, 0,,... Sequence q 3, 8, 5, 4, 35,... Sequence r 5, 0, 7,... The sequence r is obtained from sequences p and q as follows. and so on = 5, = 0, = 7 89
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