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Use of DrFrostMaths for practice
Register for free at:www.drfrostmaths.com/homework
Practise questions by chapter, including past paper Edexcel questions and extension questions (e.g. MAT).
Teachers: you can create student accounts (or students can register themselves).
Chapter Overview
Determine the value of 2 + 4 + 6 +âŻ+ 100
2:: Arithmetic Series
The first term of a geometric sequence is 3 and the second term 1. Find the sum to infinity.
3:: Geometric Series
Determine the value of
đ=1
100
(3đ + 1)
4:: Sigma Notation
If đ1 = đ and đđ+1 = 2đđ â 1, determine đ3 in terms of đ.
5:: Recurrence Relations
NEW TO A LEVEL 2017!Identifying whether a sequence is increasing, decreasing, or periodic.
1:: Sequences
Sequences
2, 5, 8, 11, 14, âŚ
+3 +3 +3 This is a:
Arithmetic Sequence?
Geometric Sequence(We will explore these later in the chapter)
?3, 6, 12, 24, 48, âŚ
Ă 2 Ă 2 Ă 2
common difference đ
common ratio đ
?
?An arithmetic sequence is one which has a common difference between terms.
A sequence is an ordered set of mathematical objects. Each element in the sequence is called a term.
Sequences
1, 1, 2, 3, 5, 8, âŚThis is the Fibonacci Sequence. The terms follow a recurrence relation because each term can be generated using the previous ones.We will encounter recurrence relations later in the chapter.
?
The fundamentals of sequences
đ˘đ The đth term. So đ˘3 would refer to the 3rd term.
đ The position of the term in the sequence.
2, 5, 8, 11, 14,âŚ
đ = 3 đ˘3 = 8
? Meaning
? Meaning
? ??
đth term of an arithmetic sequence
We use đ to denote the first term. đ is the difference between terms, and đ is the position of the term weâre interested in. Therefore:
1st Term 2nd Term 3rd Term đth term. . .
đ đ + đ đ + 2đ đ + (đ â 1)đ? ? ? ?
! đth term of arithmetic sequence:
đ˘đ = đ + đ â 1 đ
. . .
An arithmetic sequence is one which has a common difference between terms.
đth term of an arithmetic sequence
! đth term of arithmetic sequence:
đ˘đ = đ + đ â 1 đ
Example 1
The đth term of an arithmetic sequence is đ˘đ = 55 â 2đ.a) Write down the first 3 terms of the
sequence.b) Find the first term in the sequence that is
negative.
a) đđ = đđ â đ đ = đđ, đđ = đđ, đđ = đđb) đđ â đđ < đ â đ > đđ. đ â´ đ = đđđđđ = đđ â đ đđ = âđ
??
Example 2
Find the đth term of each arithmetic sequence.a) 6, 20, 34, 48, 62b) 101, 94, 87, 80, 73
a) đ = đ,đ = đđđđ = đ + đđ đ â đ= đđđ â đ
b) đ = đđđ, đ = âđđđ = đđđ â đ đ â đ= đđđ â đđ
Tip: Always write out đ =, đ =, đ = first.
?
?
Further Examples
[Textbook] A sequence is generated by the formula đ˘đ = đđ + đ where đ and đ are constants to be found.Given that đ˘3 = 5 and đ˘8 = 20, find the values of the constants đ and đ.
đ˘3 = 5 â đ + 2đ = 5đ˘8 = 20 â đ + 7đ = 20
Solving simultaneously, đ = 3, đ = â4
?
Remember that an arithmetic sequence is one where there is a common difference between terms.
đđ + đ = đđđ â đđ
đđđ â đđđ + đ = đ â đđ â đ đ â đ = đ
đ =đ
đ, đ = đ
For which values of đĽ would the expression â8, đĽ2 and 17đĽ form the first three terms of an arithmetic sequence.
?
Test Your Understanding
Edexcel C1 May 2014(R) Q10
?
?
Exercise 3A
Pearson Pure Mathematics Year 2/ASPages 61-62
Extension[STEP I 2004 Q5] The positive integers can be split into five distinct arithmetic progressions, as shown:
A: 1, 6, 11, 16, âŚB: 2, 7, 12, 17, âŚC: 3, 8, 13, 18, âŚD: 4, 9, 14, 19, âŚE: 5, 10, 15, 20, âŚ
Write down an expression for the value of the general term in each of the five progressions. Hence prove that the sum of any term in B and any term in C is a term in E.
Prove also that the square of every term in B is a term in D. State and prove a similar claim about the square of every term in C.
i) Prove that there are no positive integers đĽ and đŚ such that đĽ2 + 5đŚ = 243723
ii) Prove also that there are no positive integers đĽand đŚ such đĽ4 + 2đŚ4 = 26081974
1
A: đđ â đ B: đđ â đ C: đđ â đD: đđ â đ E: đđ
Term in B + Term in C (note that the positions in each sequence could be different, so use đ and đ):
đđ â đ + đđâ đ= đ(đ +đ â đ)
hence a term in E.
Square of term in B:
đđ â đ đ = đđđđ â đđđ + đ
= đ đđđ â đđ + đ â đ
hence a term in D.
Square of term in C:
đđ â đ đ = đđđđ â đđđ + đ
= đ đđđ â đđ + đ â đ
hence also a term in D.
i) 243723 is a term in C. Note that whatever sequence đđ is in, adding đđwill keep it in the same sequence as it is a multiple of 5. So we need to prove no term squared can give a term in C.We have already proved a term in B or in C, squared, gives a term in D. We can similarly show a term in A, squared, stays in A, and a term in E squared stays in E. Finally, a term in D squared gives a term in A. Since the sequences A, B, C, D, E clearly span all positive integers, there cannot be any integer solutions to the equation.
?
Series
A series is a sum of terms in a sequence.You will encounter âseriesâ in many places in A Level:
Arithmetic Series (this chapter!)Sum of terms in an arithmetic sequence.
2 + 5 + 8 + 11
Taylor Series (Further Maths)Expressing a function as an infinite series, consisting of polynomial terms.
đđĽ = 1 +đĽ
1!+đĽ2
2!+đĽ3
3!+âŻ
Extra Notes: A âseriesâ usually refers to an infinitesum of terms in a sequence. If we were just summing some finite number of them, we call this a partial sum of the series.
e.g. The âHarmonic Seriesâ is
1 +1
2+
1
3+
1
4+⯠, which is
infinitely many terms. But a we could get a partial sum,
e.g. đť3 = 1 +1
2+
1
3=
11
6
However, in this syllabus, the term âseriesâ is used to mean either a finite or infinite addition of terms.
Binomial Series (Later in Year 2)You did Binomial expansions in Year 1. But when the power is negative or fractional, we end up with an infinite series.
1 + đĽ = 1 + đĽ12 = 1 +
1
2đĽ â
1
8đĽ2 +
1
6đĽ3 ââŻ
Terminology: A âpower seriesâ is an infinite polynomial with increasing powers of đĽ. There is also a chapter on power series in the Further Stats module.
https://www.youtube.com/watch?v=Prc7h8lyDXg
Arithmetic Series
đ˘đ = đ + đ â 1 đ đđ =đ
22đ + đ â 1 đ
đth term ! Sum of first đ terms
?
Letâs prove it!
đđ = đ + đ + đ + đ + 2đ +⯠+ đ + đ â 1 đđđ = đ + đ â 1 đ + + đ + 2đ + đ + đ + đAdding:2đđ = 2đ + đ â 1 đ +âŻ+ 2đ + đ â 1 đ = đ 2đ + đ â 1 đ
â´ đđ =đ
22đ + đ â 1 đ
Example:
Take an arithmetic sequence 2, 5, 8, 11, 14, 17,âŚđ5 = 2 + 5 + 8 + 11 + 14
Reversing:đ5 = 14 + 11 + 8 + 5 + 2
Adding these:2đ5 = 16 + 16 + 16 + 16 + 16 = 16 Ă 5 = 80â´ đ5 = 40
Proving more generally:
The idea is that each pair of terms, at symmetrically opposite ends, adds to the same number.
Exam Note: The proof has been an exam question before. Itâs also a university interview favourite!
Alternative Formula
đ + đ + đ +âŻ+ đż
Suppose last term was đż.We saw earlier that each opposite pair of terms (first and last, second and second last, etc.) added to the same total, in this case đ + đż.
There are đ
2pairs, therefore:
!
đđ =đ
2đ + đż
Examples
Find the sum of the first 30 terms of the following arithmetic sequencesâŚ
1 2 + 5 + 8 + 11 + 14⌠đ30 = 1365
100 + 98 + 96 +⯠đ30 = 2130
đ + 2đ + 3đ +⯠đ30 = 465đ
2
3
?
?
?
Tips: Again, explicitly write out "đ =âŻ, đ = ⯠, đ = âŻâ. Youâre less likely to make incorrect substitutions into the formula.
Make sure you write đđ = ⯠so you make clear to yourself (and the examiner) that youâre finding the sum of the first đ terms, not the đth term.
đşđ > đđđđ, đ = đ, đ = đđ
đđđ + đ â đ đ > đđđđ
đ
đđ + đ â đ đ > đđđđ
đđđ + đđ â đđđđ > đđ < âđđ. đ đđ đ > đđ. đ
So 28 terms needed.
?
Find the greatest number of terms for the sum of 4 + 9 + 14 +⯠to exceed 2000.
Edexcel C1 Jan 2012 Q9
đ = 400
đ = ÂŁ24450
?
?
Test Your Understanding
Exercise 3BPearson Pure Mathematics Year 2/ASPages 64-66
[MAT 2008 1I]The function đ đ is defined for positive integers đ by
đ đ = sum of digits of đFor example, đ 723 = 7 + 2 + 3 = 12.The sum
đ 1 + đ 2 + đ 3 +âŻ+ đ 99equals what?
Extension
1
3
[MAT 2007 1J]The inequality
đ + 1 + đ4 + 2 + đ9 + 3 + âŻ+ đ10000 + 100 > đ
Is true for all đ ⼠1. It follows thatA) đ < 1300B) đ2 < 101C) đ ⼠10110000
D) đ < 5150
đ will be largest when đ is its smallest, so let đ = đ.Each of the đ terms are therefore 1, giving the LHS:
đđđ+ đ+ đ + đ + âŻ+ đđđ
đđđ+đđđ
đđ + đđ Ă đ = đđđđ
The answer is D.
?
[AEA 2010 Q2]The sum of the first đ terms of an arithmetic series is đ and the sum of the first đ terms of the same arithmetic series is đ, where đ and đ are positive integers and đ â đ.Giving simplified answers in terms of đ and đ, finda) The common difference of the terms in this
series,b) The first term of the series,c) The sum of the first đ + đ terms of the
series.
Solution on next slide.
When we sum all digits, we can separately consider sum of all units digits, and the sum of all tens digits.Each of 1 to 9 occurs ten times as units digit, so sum is đđ Ă đ+ đ+ âŻ+ đ = đđđSimilarly each of 1 to 9 occurs ten times as tens digit, thus total is đđđ + đđđ = đđđ.
2
?
Solution to Extension Q2
[AEA 2010 Q2]The sum of the first đ terms of an arithmetic series is đ and the sum of the first đ terms of the same arithmetic series is đ, where đ and đ are positive integers and đ â đ.Giving simplified answers in terms of đ and đ, finda) The common different of the terms in this series,b) The first term of the series,c) The sum of the first đ + đ terms of the series.
Recap of Arithmetic vs Geometric Sequences
2, 5, 8, 11, 14, âŚ
+3 +3 +3 This is a:
Arithmetic Sequence?
Geometric Sequence?3, 6, 12, 24, 48, âŚ
Ă 2 Ă 2 Ă 2
common difference đ
common ratio đ
?
?
! A geometric sequence is one in which there is a common ratio between terms.
Quickfire Common Ratio
Identify the common ratio đ:
1, 2, 4, 8, 16, 32,⌠đ = 21
27, 18, 12, 8,⌠đ = 2/32
10, 5, 2.5, 1.25,⌠đ = 1/23
5,â5, 5, â5, 5, â5,⌠đ = â14
đĽ,â2đĽ2, 4đĽ3 đ = â2đĽ5
1, đ, đ2, đ3, ⌠đ = đ6
4,â1, 0.25,â0.0625,⌠đ = â0.257
?
?
?
?
?
?
?
An alternating sequence is one which oscillates between positive and negative.
đth term
Arithmetic Sequence Geometric Sequence
đ˘đ = đ + đ â 1 đ đ˘đ = đđđâ1
Determine the 10th and đth terms of the following:
3, 6, 12, 24, âŚ
40, -20, 10, -5, âŚ
? ?
đ = 3, đ = 2đ˘10 = 3 Ă 29 = 1536đ˘đ = 3 2đâ1
đ = 40, đ = â1
2
đ˘10 = 40 Ă â1
2
9
= â5
64
đ˘đ = â1 đâ1 Ă5
2đâ4
Tip: As before, write out đ =and đ = first before substituting.
?
?
Further Example
[Textbook] The second term of a geometric sequence is 4 and the 4th
term is 8. The common ratio is positive. Find the exact values of:a) The common ratio.b) The first term.c) The 10th term.
đ˘2 = 4 â đđ = 4 (1)đ˘4 = 8 â đđ3 = 8 (2)
a) Dividing (2) by (1) gives đ2 = 2, đ đ đ = 2
b) Substituting, đ =4
đ=
4
2= 2 2
c) đ˘10 = đđ9 = 64
Tip: Explicitly writing đ˘2 = 4 first helps you avoid confusing the đth term with the âsum of the first đ termsâ (the latter of which weâll get onto).
?
Further Example
[Textbook] The numbers 3, đĽ and đĽ + 6 form the first three terms of a positive geometric sequence. Find:
a) The value of đĽ.b) The 10th term in the sequence.
đĽ
3=đĽ + 6
đĽâ đĽ2 = 3đĽ + 18
đĽ2 â 3đĽ â 18 = 0đĽ + 3 â6 = 0đĽ = 6 đđ â 3But there are no negative terms so đĽ = 6
đ =đĽ
3=6
3= 2 đ = 3 đ = 10
đ˘10 = 3 Ă 29 = 1536
?
Hint: Youâre told itâs a geometric sequence, which means the ratio between successive terms must be the
same. Consequently đ˘2
đ˘1=
đ˘3
đ˘2
Exam Note: This kind of question has appeared in the exam multiple times.
đth term with inequalities
[Textbook] What is the first term in the geometric progression 3, 6, 12, 24,⌠to exceed 1 million?
đ˘đ > 1000000 đ = 3, đ = 2â´ 3 Ă 2đâ1 > 1000000
2đâ1 >1000000
3
log 2đâ1 > log1000000
3
đ â 1 >log
10000003
đđđ2đ â 1 > 18.35đ > 19.35đ = 20
?
Test Your Understanding
All the terms in a geometric sequence are positive.The third term of the sequence is 20 and the fifth term 80. What is the 20th term?
đ˘3 = 20 â đđ2 = 20đ˘5 = 80 â đđ4 = 80
Dividing: đ2 = 4, â´ đ = 2
đ =20
đ2=20
4= 5
đ˘20 = 5 Ă 219 = 2621440
The second, third and fourth term of a geometric sequence are the following:đĽ, đĽ + 6, 5đĽ â 6
a) Determine the possible values of đĽ.b) Given the common ratio is positive, find the common ratio.c) Hence determine the possible values for the first term of the sequence.
đĽ + 6
đĽ=5đĽ â 6
đĽ + 6đĽ + 6 2 = đĽ 5đĽ â 6đĽ2 + 12đĽ + 36 = 5đĽ2 â 6đĽ4đĽ2 â 18đĽ â 36 = 02đĽ2 â 9đĽ â 18 = 02đĽ + 3 đĽ â 6
đĽ = â3
2đđ đĽ = 6
đ =đĽ + 6
đĽ=6 + 6
6= 2
đ˘2 = đĽ = 6đđ = 6
đ =6
2= 3
?
?
Exercise 3C
Pearson Pure Mathematics Year 2/ASPages 69-70
Sum of the first đ terms of a geometric series
Arithmetic Series Geometric Series
đđ =đ
22đ + đ â 1 đ đđ =
đ 1 â đđ
1 â đ? ?
Proof: đđ = đ + đđ + đđ2 +âŻ+ đđđâ2 + đđđâ1
Mutiplying by đ:đđđ = đđ + đđ2 +⯠+ đđđâ1 + đđđ
Subtracting:đđ â đđđ = đ â đđđ
đđ 1 â đ = đ 1 â đđ
đđ =đ 1 â đđ
1 â đ
?
Exam Note: This once came up in an exam. And again is a university interview favourite!
Examples
đđ =đ 1 â đđ
1 â đ
3, 6, 12, 24, 48,âŚ
Find the sum of the first 10 terms.
đ = 3, đ = 2, đ = 10
đđ =3 1 â 210
1 â 2= 3069
4, 2,1,1
2,1
4,1
8, ⌠đ = 4, đ =
1
2, đ = 10
đđ =
4 1 â12
10
1 â12
=1023
128
? ? ?
?
?
? ? ?
Geometric Series
Harder Example
Find the least value of đ such that the sum of 1 + 2 + 4 + 8 +âŻto đ terms would exceed 2 000 000.
đ = 1, đ = 2, đ =?đđ > 2 000 000
1 1 â 2đ
1 â 2> 2 000 000
2đ â 1 > 2 000 0002đ > 2 000 001đ > log2 2000001đ > 20.9
So 21 terms needed.
đđđ2 both sides to cancel out â2 to the power ofâ.
?
?
Test Your Understanding
Edexcel C2 June 2011 Q6
?
?
?
Exercise 3D
Pearson Pure Mathematics Year 2/ASPages 72-73
Extension
[MAT 2010 1B]The sum of the first 2đ terms of
1, 1, 2,1
2, 4,
1
4, 8,
1
8, 16,
1
16, âŚ
is
A) 2đ + 1 â 21âđ
B) 2đ + 2âđ
C) 22đ â 23â2đ
D)2đâ2âđ
3
1 There are interleaved sequences. If we want đđterms, we want đ terms of đ, đ, đ, đ, ⌠and đ
terms of đ,đ
đ,đ
đ, âŚ
For first sequence:
đşđ =đ đ â đđ
đ â đ= đđ â đ
For second sequence:
đşđ =đ đ â
đđ
đ
đ âđđ
=đ â đâđ
đ/đ= đ â đđâđ
Therefore the sum of both is (A).
?
Divergent vs Convergent
1 + 2 + 4 + 8 + 16 + ...
What can you say about the sum of each series up to infinity?
1 â 2 + 3 â 4 + 5 â 6 + âŚ
1 + 0.5 + 0.25 + 0.125 + ...
1
1+1
2+1
3+1
4+âŻ
This is divergent â the sum of the values tends towards infinity.
This is divergent â the running total alternates either side of 0, but gradually gets further away from 0.
This is convergent â the sum of the values tends towards a fixed value, in this case 2.
This is divergent . This is known as the Harmonic Series
1
1+1
4+1
9+
1
16+âŻ
This is convergent . This is known as the Basel Problem, and the value is đ2/6.
Definitely NOT in the A Level syllabus, and just for fun...
?
?
?
?
?
Sum to Infinity
1 + 0.5 + 0.25 + 0.125 + ...
1 + 2 + 4 + 8 + 16 + ...
Why did this infinite sum converge (to 2)âŚ
âŚbut this diverge to infinity?
The infinite series will converge provided that â1 < đ < 1 (which can be written as đ < 1), because the terms will get smaller.
Provided that đ < 1, what happens to đđ as đ â â?
For example đ
đ
đđđđđđis very close to 0.
We can see that as đ â â, đđ â đ.
How therefore can we use the đđ =đ 1âđđ
1âđformula
to find the sum to infinity, i.e. đâ?
! For a convergent geometric series,
đâ =đ
1 â đ
?
! A geometric series is convergent if đ < 1.
Quickfire Examples
1,1
2,1
4,1
8, ⌠đ = đ, đ =
đ
đđşâ = đ
27,â9,3,â1,⌠đ = đđ, đ = âđ
đđşâ =
đđ
đ
đ, đ2, đ3, đ4, ⌠đ = đ, đ = đ đşâ =đ
đ â đđ¤âđđđ â 1 < đ < 1
? ?
? ?
?
??
?
?
đ, 1,1
đ,1
đ2, ⌠đ = đ, đ =
đ
đđşâ =
đđ
đ â đ???
Further Examples
[Textbook] The fourth term of a geometric series is 1.08 and the seventh term is 0.23328.a) Show that this series is convergent.b) Find the sum to infinity of this
series.
đ˘4 = 1.08 â đđ3 = 1.08đ˘7 = 0.23328 â đđ6 = 0.23328
Dividing:đ3 = 0.216 â´ đ = 0.6
The series converges because 0.6 < 1.
đ =1.08
đ3=
1.08
0.216= 5
â´ đâ =5
1 â 0.6= 12.5
? a
? b
[Textbook] For a geometric series with first term đ and common ratio đ, đ4 =15 and đâ = 16.a) Find the possible values of đ.b) Given that all the terms in the series are positive, find the value of đ.
đ4 = 15 âđ 1 â đ4
1 â đ= 15 (1)
đâ = 16 âđ
1 â đ= 16 (2)
Substituting đ
1âđfor 16 in equation (1):
16 1 â đ4 = 15
Solving: đ = Âą1
2
As terms positive, đ =1
2, â´ đ = 16 1 â
1
2= 8
Fro Warning: The power is đ in the đđ formula but đ â 1 in the đ˘đ formula.
? a
? b
Test Your Understanding
Edexcel C2 May 2011 Q6
đ =đ
đ
đ = đđđ
đşâ = đđđđ
đđđ đ âđđ
đ
đ. đđ> đđđđ
đ >đđđ
đđđđ
đđđ đ. đđâ đ = đđ
?
?
?
?
Exercise 3E
Pearson Pure Mathematics Year 2/ASPages 75-76
Extension
[MAT 2006 1H] How many solutions does the equation2 = sin đĽ + sin2 đĽ + sin3 đĽ + sin4 đĽ + âŻ
have in the range 0 ⤠đĽ < 2đ
RHS is geometric series. đ = đŹđ˘đ§ đ , đ = đŹđ˘đ§ đ.
đ =đŹđ˘đ§ đ
đ â đŹđ˘đ§ đâ đ â đđŹđ˘đ§ đ = đŹđ˘đ§ đ
đŹđ˘đ§ đ =đ
đBy considering the graph of sin for đ ⤠đ < đđ , this has 2 solutions.
1[MAT 2003 1F] Two players take turns to throw a fair six-sided die until one of them scores a six. What is the probability that the first player to throw the die is the first to score a six?
đˇ đđđ đđđ đđđđđ =đ
đ
đˇ đđđ đđđ đđđđđ =đ
đ
đ
Ăđ
đ
đˇ đđđ đđđ đđđđđ =đ
đ
đ
Ăđ
đ
The total probability of the 1st player winning is the sum of these up to infinity.
đ =đ
đ, đ =
đ
đ
đ
=đđ
đđ
đşâ =đ/đ
đ â đđ/đđ=
đ
đđ
2
[Frost] Determine the value of đĽ where:
đĽ =1
1+2
2+3
4+4
8+
5
16+
6
32+âŻ
(Hint: Use an approach similar to proof of geometric đđ formula)Doubling:
đđ = đ +đ
đ+đ
đ+đ
đ+đ
đ+âŻ
Subtracting the two equations:
đ = đ +đ
đ+đ
đ+đ
đ+đ
đ+âŻ
= đ + đ = đ
N
?
?
?
Sigma Notation
đ=1
5
(2đ + 1)
What does each bit of this expression mean?
The Greek letter, capital sigma, means âsumâ.
The numbers top and bottom tells us what đ varies between. It goes up by 1 each time.
We work out this expression for each value of đ (between 1 and 5), and add them together.
= 3
đ = 1
+5
đ = 2
+7
đ = 3
+9
đ = 4
+11
đ = 5
= 35If the expression being summed (in this case 2đ + 1) is linear, we get an arithmetic series. We can therefore apply our usual approach of establishing đ, đ and đ before applying the đđ formula.
Determining the value
đ=1
7
3đ
đ=5
15
10 â 2đ
First few terms? Values of đ, đ, đ đđ đ?
= 3 + 6 + 9 +⯠đ = 3, đ = 3, đ = 7
Final result?
đ7 =7
26 + 6 Ă 3
= 84
= 0 + â2 + â4 +⯠đ = 0, đ = â2, đ = 11Be careful, there are 11 numbers between 5 and 15 inclusive. Subtract and +1.
đ11 =11
20 + 10 Ă â2
= â110
đ=1
12
5 Ă 3đâ1 = 5 + 15 + 45 +⯠đ = 5, đ = 3, đ = 12đ12 =
5 1 â 312
1 â 3= 1328600
đ=5
12
5 Ă 3đâ1 Note: You can either find from scratch by finding the first few terms (the first when đ = 5), or by calculating:
đ=1
12
5 Ă 3đâ1 â
đ=1
4
5 Ă 3đâ1
i.e. We start with the first 12 terms, and subtract the first 4 terms.
? ? ?
? ? ?
? ? ?
?
Solomon Paper A
Testing Your Understanding
Method 1: Direct
đ=10
30
7 + 2đ =27 + 29 + 31 +âŻ
đ = 27, đ = 2, đ = 21
â´ đ21 =21
254 + 20 Ă 2
= 987
Method 2: Subtraction
đ=10
30
7 + 2đ =
đ=1
30
7 + 2đ â
đ=1
9
7 + 2đ
đ=1
30
7 + 2đ = 9 + 11 + 13 +âŻ
đ = 9, đ = 2, đ = 30
đ30 =30
218 + 29 Ă 2 = 1140
đ9 =9
218 + 8 Ă 2 = 153
1140 â 153 = 987
?
?
On your calculator
âUse of Technologyâ Monkey says:The Classwiz and Casio Silver calculator has a ÎŁ button.
Try and use it to find:
đ=5
12
2 Ă 3đ
Exercise 3F
Pearson Pure Mathematics Year 2/ASPages 77-78
Recurrence Relations
đ˘đ = 2đ2 + 3This is an example of a position-to-term sequence, because each term is based on the position đ.
đ˘đ+1 = 2đ˘đ + 4đ˘1 = 3
But a term might be defined based on previous terms.If đđ refers to the current term, đđ+đrefers to the next term.So the example in words says âthe next term is twice the previous term + 4â
This is known as a term-to-term sequence, or more formally as a recurrence relation, as the sequence ârecursivelyâ refers to itself.
We need the first term because the recurrence relation alone is not enough to know what number the sequence starts at.
Example
Important Note: With recurrence relation questions, the the sequence will likely not be arithmetic nor geometric. So your previous đ˘đ and đđ formulae do not apply.
Edexcel C1 May 2013 (R)
đĽ2 = đĽ12 â đđĽ1 = 1 â đ
đ =3
2
đĽ3 = đĽ22 â đđĽ2
= 1 â đ 2 â đ 1 â đ= 1 â 3đ + 2đ2
= 1 + â1
2+ 1 + â
1
2+ âŻ
= 50 Ă 1 â1
2= 25
?
?
?
?Tip: When a ÎŁ question comes up for recurrence relations, it will most likely be some kind of repeating sequence. Just work out the sum of terms in each repeating bit, and how many times it repeats.
Test Your Understanding
Edexcel C1 Jan 2012
đĽ2 = đ + 5
đĽ3 = đ đ + 5 + 5 = âŻ
đ2 + 5đ + 5 = 41đ2 + 5đ â 36 = 0đ + 9 đ â 4 = 0đ = â9 đđ 4
?
?
?
Combined Sequences
Sequences (or series) can be generated from a combination of both an arithmetic and a geometric sequence.
Example
Exercise 3G
Pearson Pure Mathematics Year 2/ASPages 80-81
[AEA 2011 Q3] A sequence {đ˘đ} is given byđ˘1 = đđ˘2đ = đ˘2đâ1 Ă đ, đ ⼠1đ˘2đ+1 = đ˘2đ Ă đ đ ⼠1
(a) Write down the first 6 terms in the sequence.
(b) Show that Ďđ=12đ đ˘đ =
đ 1+đ 1â đđ đ
1âđđ
(c) [đĽ] means the integer part of đĽ, for example 2.73 = 2, 4 = 4.
Find Ďđ=1â 6 Ă
4
3
đ
2 Ă3
5
đâ1
2
[MAT 2014 1H] The function đš đ is defined for all positive integers as follows: đš 1 = 0 and for all đ ⼠2,đš đ = đš đ â 1 + 2 if 2 divides đ but 3 does not divide ,đš đ = đš đ â 1 + 3 if 3 divides đ but 2 does not divide đ,đš đ = đš đ â 1 + 4 if 2 and 3 both divide đđš đ = đš đ â 1 if neither 2 nor 3 divides đ.Then the value of đš 6000 equals what?Solution: 11000
[MAT 2016 1G] The sequence đĽđ , where đ ⼠0, is defined by đĽ0 = 1 and
đĽđ = Ďđ=0đâ1 đĽđ for đ ⼠1
Determine the value of the sum
đ=0
â 1
đĽđSolution on next slide.
1
2 3
?
?
Solution to Extension Question 3
[MAT 2016 1G] The sequence đĽđ , where đ ⼠0, is defined by đĽ0 = 1 and
đĽđ = Ďđ=0đâ1 đĽđ for đ ⼠1
Determine the value of the sum
đ=0
â 1
đĽđ
Solution:đđ = đđđ = đđđ = đ + đ = đđđ = đ + đ + đ = đ
This is a geometric sequence from đđ onwards.
Therefore
đ=đ
â đ
đđ=đ
đ+đ
đ+đ
đ+đ
đ+đ
đ+âŻ
= đ + đ = đ
A somewhat esoteric Futurama joke explained
This simplifies to
đ = đ0
đ=1
â1
đ
The sum 1
1+
1
2+
1
3+⯠is known as the harmonic series, which is divergent.
Bender (the robot) manages to self-clone himself, where some excess is required to produce the duplicates (e.g. alcohol), but the duplicates are smaller versions of himself. These smaller clones also have the capacity to clone themselves. The Professor is worried that the total amount mass đ consumed by the growing population is divergent, and hence theyâll consume to Earthâs entire resources.
Increasing, decreasing and periodic sequences
A sequence is strictly increasing if the terms are always increasing, i.e.đ˘đ+1 > đ˘đ for all đ â â.e.g. 1, 2, 4, 8, 16, âŚ
Similarly a sequence is strictly decreasing if đ˘đ+1 < đ˘đ for all đ â â
A sequence is periodic if the terms repeat in a cycle. The order đ of a sequence is how often it repeats, i.e. đ˘đ+đ = đ˘đ for all đ.e.g. 2, 3, 0, 2, 3, 0, 2, 3, 0, 2, ⌠is periodic and has order 3.
[Textbook] For each sequence:i) State whether the sequence is increasing, decreasing or periodic.ii) If the sequence is periodic, write down its order.
a) đ˘đ+1 = đ˘đ + 3, đ˘1 = 7 Terms: 7, 10, 13,âŚ. Increasing.
b) đ˘đ+1 = đ˘đ2, đ˘1 =
1
2Terms:
1
2,1
4,1
8,1
16, âŚ. Decreasing.
c) đ˘đ+1 = sin 90đ° Terms: 1, 0, -1, 0, 1, 0, ⌠Periodic, order 4.
???
Textbook Error: It uses the term âincreasingâ when it means âstrictly increasingâ.
Exercise 3H
Pearson Pure Mathematics Year 2/ASPages 82-83
Modelling
Anything involving compound changes (e.g. bank interest) will form a geometric sequence, as there is a constant ratio between terms.We can therefore use formulae such as đđ to solve problems.
[Textbook] Bruce starts a new company. In year 1 his profits will be ÂŁ20 000. He predicts his profits to increase by ÂŁ5000 each year, so that his profits in year 2 are modelled to be ÂŁ25 000, in year 3, ÂŁ30 000 and so on. He predicts this will continue until he reaches annual profits of ÂŁ100 000. He then models his annual profits to remain at ÂŁ100 000.a) Calculate the profits for Bruceâs business in the first 20 years.b) State one reason why this may not be a suitable model.c) Bruceâs financial advisor says the yearly profits are likely to increase by 5% per annum.
Using this model, calculate the profits for Bruceâs business in the first 20 years.
The sequence is arithmetic up until ÂŁ100 000, but stops increasing thereafter. We need to know the position of this term.
đ = đđ đđđ, đ = đđđđđđ = đ + đ â đ đ â đđđđđđ= đđđđđ + đ â đ đ â´ đ = đđ
First find sum of first 17 terms (where sequence is arithmetic):
đşđđ =đđ
đ(đ đđđđđ + đđ đđđđ = đ đđđ đđđ
â´ đşđđ = đ đđđ đđđ + đ đđđ đđđ = đ đđđ đđđ
? a
It is unlikely that Bruceâs profits will increase by exactly the same amount each year.
đ = 20 000 đ = 1.05
đ20 =20000 1.0520 â 1
1.05 â 1= ÂŁ 661 319.08
? b
? c
a b
c
Geometric Modelling Example
[Textbook] A piece of A4 paper is folded in half repeatedly. The thickness of the A4 paper is 0.5 mm.(a) Work out the thickness of the paper after four folds.(b) Work out the thickness of the paper after 20 folds.(c) State one reason why this might be an unrealistic model.
đ = 0.5 đđ, đ = 2After 4 folds:đ˘5 = 0.5 Ă 24 = 8 đđ
After 20 foldsđ˘21 = 0.5 Ă 220 = 524 288 đđ
It is impossible to fold the paper that many times so the model is unrealistic.
a
b
c
?
?
?
Test Your Understanding
Edexcel C2 Jan 2013 Q3
? a
? b
? c
Exercise 3I
Pearson Pure Mathematics Year 2/ASPages 84-86
[AEA 2007 Q5]
The figure shows part of a sequence đ1, đ2, đ3, âŚ, of model snowflakes. The first term đ1 consist of a single square of side đ. To obtain đ2, the middle third of each edge is replaced with a new
square, of side đ
3, as shown. Subsequent terms are added by replacing
the middle third of each external edge of a new square formed in the
previous snowflake, by a square 1
3of the size, as illustrated by đ3.
a) Deduce that to form đ4, 36 new squares of side đ
27must be added
to đ3.
b) Show that the perimeters of đ2 and đ3 are 20đ
3and
28đ
3respectively.
c) Find the perimeter of đđ.
1
Extension
?
Just for your interestâŚ
Tell me about this âharmonic seriesââŚ
A harmonic series is the infinite summation:
1 +1
2+1
3+1
4+ âŻ
We can also use đťđ to denote the sum up to the đth term (known as a partial sum), i.e.
đťđ = 1 +1
2+1
3+ âŻ+
1
đ
Is it convergent or divergent?The terms we are adding gradually get smaller, so we might wonder
if the series âconvergesâ to an upper limit (just like 1 +1
2+
1
4+
1
8+âŻ
approaches/converges to 2), or is infinitely large (i.e. âdivergesâ).
We can prove the series diverges by finding a value which is smaller, but that itself diverges:
đť = 1 +1
2+1
3+1
4+1
5+1
6+1
7+1
8+1
9+âŻ
đĽ = 1 +1
2+1
4+1
4+1
8+1
8+1
8+1
8+
1
16+ âŻ
Above we defined a new series đĽ, such that we
use one 1
2, two
1
4đ , four
1
8đ , eight
1
16s and so on.
Each term is clearly less than (or equal to) each term in đť. But it simplifies to:
đĽ = 1 +1
2+1
2+1
2+1
2+âŻ
which diverges. Since đĽ < đť, đť must also diverge.
It is also possible to prove that đŻđ is never an integer for any đ > đ.
Where does the name come from?The âharmonic meanâ is used to find the average of rates. For example, if a cat runs to a tree at 10mph and back at 20mph, its average speed across
the whole journey is 131
3mph (not 15mph!).
Each term in the harmonic series is the harmonic mean of the two
adjacent terms, e.g. 1
3is the harmonic mean of
1
2and
1
4. The actual word
âharmonicâ comes from music/physics and is to do with sound waves.
10đđâ
20đđâ
The harmonic mean
of đĽ and đŚ is 2đĽđŚ
đĽ+đŚ.
đđđđđđđđđđ
DrFrostMaths.com
The time Dr Frost solved a maths problem forMy mum, who worked at John Lewis, was selling London Olympics âtrading cardsâ, of which there were about 200 different cards to collect, and could be bought in packs. Her manager was curious how many cards you would have to buy on average before you collected them all. The problem was passed on to meâŚ
It is known as the âcoupon collectorâs problemâ.The key is dividing purchasing into separate stages where each stage involves getting the next unseen cardThere is an initial probability of 1 that a purchase results in an
unseen card. There is then a 199
200chance the next purchase
results in an unseen card. Reciprocating this probability gives
us the number of cards on average, i.e. 200
199, weâd have to buy
until we get our second unseen card. Eventually, there is a 1
200
chance of a purchase giving us the last unseen card, for which
weâd have to purchase 200
1= 200 cards on average. The total
number of cards on average we need to buy is therefore:200
200+200
199+ âŻ+
200
2+200
1
= 2001
200+
1
199+ âŻ+
1
2+ 1
= 200 Ă đť200 = 1176 đđđđđ
Youâre welcome John Lewis.
Really cool applications: How far can a stack of đbooks protrude over the edge of a table?
1 book: 1
2a length.
2 books: 1
2+
1
4=
3
4of a length.
1
2
3
4
3 books: 1
2+
1
4+
1
6=
11
12
11
12
Each of these additions is half the sum of the first đ terms of the harmonic series.
e.g. 3 books: 1
21 +
1
2+
1
3=
11
12
The overhang for đ books is therefore đ
đđŻđ
Because the harmonic series is divergent, we can keep adding books to get an arbitrarily large amount of overhang. But note that the harmonic series increases very slowly. A stack of 10,000 books would only overhang by 5 book lengths!