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8/3/2019 Maths SL Algebra
1/11
5. Algebra
Arithmetic sequences and series
You should be able to:
state whether a sequence is arithmetic, giving an
appropriate reason
fnd the common dierence in an arithmetic sequence
fnd the nth term o an arithmetic sequence fnd the number o terms in an arithmetic sequence
calculate the sum o the frst n terms o an arithmetic
series
solve real-world problems involving arithmetic sequences.
You should know:
a sequence is arithmetic i the dierence between
consecutive terms is the same
an arithmetic sequence has the orm u1, u1 + d, u1 + 2d,
u1 + 3d, u1 + 4d, u1 = (n 1)
the common dierence can be ound by subtracting
consecutive terms: d= un + 1 un
when to use the term ormula and when to use the sum
ormula.
Example
Consider the series 17, 7, 3, 303.
(a) Show that the series is arithmetic.
Shw ha h dirnc bwn w cncuiv rm
i cnan.Fr eamp l : 7 17 = 3 7 = 10
Trr, d = 10
(b) Find the sum o the series.
T rmula r h um an a rihmic ri rquirh valu n. U h rm rmula r nd n.
Fr eampl: 17 + (n 1)(10) = 303, n = 33
Nw u h ap prpria rmula, Sn =n(u1+ un)_______2 , nd
h um h r 33 rm,
S33
= 33(17 303)________2S33 = 4,719
Be prepared
Look or words or expressions that suggest the use o
the term ormulaater the 10th month, in the 8th
rowand those that suggest the sum ormulatotal
cost, total distance, altogether.
Look or questions in which inormation is given about
two terms. This normally suggests the ormation o a pair
o simultaneous equations or which you will have to solve
to fnd the frst term and the common dierence.
The last term o a given sequence can be used to fnd the
number o terms in the sequence as demonstrated in the
example above.
8/3/2019 Maths SL Algebra
2/11
2
5.Algebra
Geometric sequences and series
You should be able to:
state whether a sequence is geometric, giving an
appropriate reason
fnd the common ratio in a geometric sequence
fnd the nth term o a geometric sequence
fnd the number o terms in a geometric sequence
calculate the sum o the frst n terms o a geometric series
solve real-world problems involving geometric sequences
and series.
You should know:
a sequence is geometric i the ratio o consecutive terms
is the same
a geometric sequence has the orm u1, u1r, u1r2, u1rn 1
the common ratio can be ound by dividing consecutive
terms: r=un + 1____un
when to use the term ormula and when to use the sum
ormula.
Example
A sum o $5,000 is invested at a compound interestrate o 6.3% per annum.
(a) Write down an expression or the value o the
investment ater n ull years.
I may b hlp ul wri u h qunc . Fr eampl :
u0 = 5,000
u1 = 5,000 1.063
u2 = 5,000 1.0632
S ar n ull yar, un = 5,000 1.063n.
(b) What will be the value o the investment at the end
o 5 years?
W nd nly ubiu n = 5 in ur a nwr r par(a). Trr,
u5 = 5,000 1.0635
u5 = 6,790
Noic hw h anwr ha bn givn 3 ignican gur.
(c) The value o the investment will exceed $10,000
ater n ull years.
(i) Write down an inequality to represent this
inormation.
W wuld lik knw whn un > 10,000.
Tr r, 5,000 1.063n > 10,000, r 1.063n > 2
(ii) Calculate the minimum value on.
Uing a graphical ap prach:
Casio
Texas Instruments
n > 11.345
Trr, h minimum va lu n r which hinvmn ecd $10,000 i 12.
8/3/2019 Maths SL Algebra
3/11
Geometric sequences and series (continued)
Be prepared
Always write out the frst ew terms o a sequence i they
are not given. This will help you to identiy what type
o sequence is involved and thereore which ormula is
appropriate. Questions set in the context o population growth or
compound interest oten require you to fnd the amount
o time t(or n) ater which a population or investment
exceeds a certain value. Be sure to set up an appropriate
inequality and use an algebraic or graphical method to
fnd t(or n) and then interpret your answer correctly!
Knowing how to use the sequence mode on your
calculator to generate and display the terms o a sequence
can be helpul.
Casio
Texas Instruments
Use the sum and sequence unctions to evaluate
expressions written in sigma notation. For example, to
observe the terms o the arithmetic sequencen = 1
6 (2n + 5)
and to evaluate its sum, we could use the ollowing:
Casio
Texas Instruments
8/3/2019 Maths SL Algebra
4/11
4
5.Algebra
The binomial theorem
You should be able to:
use the binomial theorem to expand (a + b)n
fnd a given term in the expansion o (a + b)n.
You should know:
the general orm o an expansion is (a + b)n = n0an + n1an 1b + + nran rbr+ + nnbn
the expansion o (a + b)n:
the number o terms is always n + 1
the frst term is an and the last term is bn
the exponent o a decreases by 1 rom term to term
the exponent o b increases by 1 rom term to term
or each term, the sum o the exponents on a and b is
equal to n
the coefcients o (a + b)n are symmetricalthat is,
they read the same way rom let to right as they dorom right to let
Pascals triangle can also be used to generate the
coefcients o (a + b)n
how to use the nCreature to fnd and/or generate
binomial coefcients:
Texas Instruments Casio
ExampleConsider the expansion o2__3x 38.(a) Write down the number o terms in the expansion.
A n = 8, h numbr rm in h epa nin mu b 9.
(b) Find the term inx3.
Making u h rmula r h gnral rm a
binmial epanin, w hav a = 2_3, b = 3 and n = 8.Trr, 8
r2_3x8 r(3)r.
Fr h rm in x3, w rquir ha r = 5.
Trr, 852_3x8 5(3)5 = 4,032x3Chck: Hw d w knw ha ur an wr huld b
ngaiv?
Be prepared
Learning how to use the ormula or the general term o
an expansion correctly can save valuable time on an exam
paper. Writing out the entire expansion is not normally the
most efcient approach.
Using brackets correctly when writing down the parts o
a term will ensure that the sign o your answer is correct.
Remember that in the expansion o (a + b)n, the signs on
each term always alternate.
8/3/2019 Maths SL Algebra
5/11
Exponents and logarithms
You should be able to:
apply the laws o exponents to simpliy expressions
solve exponential equations by making the bases the
same
use logarithms to solve exponential equations with
dierent bases
apply the laws o logarithms to simpliy expressions
fnd a logarithm to any base by using the change o base
ormula
solve equations involving logarithms.
You should know:
the basic laws o exponents
how to think o logs as exponentsthat is, logaxmeans,the exponent to which a must be raised to givex. For
example, log3 9 means the exponent to which 3 must be
raised to give 9, which is 2. Thereore, log3 9 = 2
the laws o logarithms:
logaM + logaN = logaMN
logaM logaN = logaM__N
logaMn = n logaM
an exponential equation has a variable in the exponent,
or example, 2x= 7
you can solve an exponential equation by using theproperties o exponents to make the bases the same.
I the bases are the same, then the exponents must be
equal. For example:
9x 1 = 1__32x 32(x 1) = 32x 2(x1) = 2xx= 1__2 i it is not possible to make the bases the same, you can
use the properties o logarithms and the change o base
ormula to solve an exponential equation. For example:
2x= 3xln 2 = ln 3x 1.58
Example
Given that log5x =y, express each o the ollowing in
terms oy.
(a) log5x2.
Uing h hird law abv,
lg5x2= 2 lg5x
= 2y
(b) log51__xW can u h law epnn and lgarihm wrihi eprsin a llw:
lg51_x = lg5x1= lg5x
= y
(c) log25x
T chan g ba rmula m ak hi quin a nap!Uing ba 5 lgarihm, w hav
lg25x =lg5x____lg525
=y_2Be prepared
One o the keys to success in solving logarithmic equations
is your ability to move easily between the logarithmic orm
and the exponential orm. Remember that i a = logbx,
thenx= ba.
The change o base ormula allows you to express any
logarithm in terms o another base. This is useul when
graphing logarithmic unctions or evaluating logarithms
that do not have base 10 or base e.
8/3/2019 Maths SL Algebra
6/11
6
5.Algebra
Infnite geometric series
You should be able to:
state the conditions on the common ratio or which the
sum o an infnite geometric series exists
fnd the sum, i it exists, o an infnite geometric series.
You should know:
i the common ratio has a value between 1 and 1, the
series will converge to an infnite sum.
i the common ratio has a value less than 1 or greater
than 1, |r| > 1, the series will diverge and the sum to
infnity is not defned.
Example
Find the sum to infnity o the geometric series
2__3 4__9 + 8___27 16___81 + A h qunc alrna bwn piiv andngaiv valu, h cmmn rai mu b ngaiv.
T pat rn ugg ha un + 1 = 2
_3un r = 2_3an dh um inni y ei.
S =2_3
_____1 ( 2_3)
= 2_5Be prepared Beore you use the ormula or an infnite geometric
series, be sure that your common ratio is a number
between 1 and 1. I you are asked or an infnite sum
and your common ratio is greater than 1 or less than
1, you have made a mistake in an earlier part o the
question.
Sigma notation
You should be able to:
express a series using sigma notation
expand and evaluate an expression written in sigma
notation.
You should know:
the Greek letter is used to denote the sum o the
terms o a series
to express a series using sigma notation, use the orm
k= 1
n (generalterm). For example,
k= 1
5 32 1__4k 1.Example
(a) Write down the frst three terms o the sequence
un = 3
n,n
1.Subiu ing n = 1, 2 and 3 in h eprsin r un,w bain u1 = 3, u2 = 6, u3 = 9.
(b) Find,
(i) n = 1
203n
T anw r par (a) ll u h ri iarihm ic, hi quin i aking u nd hum h r 20 rm h arihmic ri.
S20=20(3+ 60)______
2
= 630
Yur GDC culd al hav bn ud nd hi um.
(ii) n = 21
1003n
Rmm br hi i a nd quin and w a r akd nd h um h rm fm n = 21 n = 100.W culd n d h um h r 100 rm and hnubrac ur a nwr pa r (b)(i).
n = 21
100 3n= S100 S20
= 100(3 + 300)_________2 630= 14,520Be prepared
Both arithmetic and geometric series can be expressed
using sigma notation.
8/3/2019 Maths SL Algebra
7/11
1. [Maximum mark: 5]
Consider the infnite geometric sequence 3 3 0 3 0 3 0 3
, ( . ) , ( . ) , ( . ) , .
(a) Write down the 10th term of the sequence. Do not simplify your answer. [1 mark]
(b) Find the sum of the infnite sequence. [4 marks]
[Taken from paper 1 November 2008]
What are the key areas rom
the syllabus?
Geometricsequences
Sumofinnitegeometricseries
How do I approach the question?
(a) As this is rom paper 1 and thereore does not allow a GDC, it is important to
know how a geometric sequence is ormedthat the tenth term is ound by
multiplying the frst term by the common ratio nine times. Once you identiythe frst term and the common ratio, consider the ormula or the nth term
o a geometric sequence. No working is needed as the command term write
down indicates that a correct answer is all that is required or the 1 mark.
(b) The question asks or the infnite sum, so be sure you choose the correct
ormula rom the inormation booklet. As you are asked to fnd this sum,
you need to show some working. Marks are earned when you substitute
correct values into the ormula and calculate the fnal result.
This answer achieved 2/5
Un= U1rn1
(a) U 1= 31 U2= 3r21= 3r1 r = 0.9
U10= U1r101= 3(0.9)9 U1= 3
U10= 3(0.9)9
A1
(b) Sn=U1( r
n 1)_______r 1
=U1(1 r
n)________1 r
Sn=3(0.9n 1)________0.9 1 = 3(1 0.9n)________1 0.9 = 3(1 0.9n)________0.1 A1 A0 A0
Sn=3 (1 0.9n)_________
0.1 A0
Unortunately, this student chosethe ormula or the fnite geometric
series. As the approach is incorrect,
no marks are earned or substituting
values into that ormula. The correct
approach would be to use the ormula or
an infnite sum,S=u1_____
1 r.
A mark is earned or recognizing
that the common ratio is 0.9.
The student wrote down a correct
tenth term, using an exponent o 9.
8/3/2019 Maths SL Algebra
8/11
8
5.Algebra
This answer achieved 4/5
(a) U n= U1+ (n 1)d
U10= 3 + (10 1)0.9
U10= 3 + 9(0.9)
(a) U 10= U1rn1
= 3 x (0.9)101
= 3 x (0.9)9
= 3(0.9)9 A1
(b) S =u1____
1 r
S =3_____
1 0.9A1 A1
S =3___
0.1A1 A0
The student chooses the correct
ormula or the sum o an infnite
geometric series, and then substitutes
correct values or u1 and r.
This student decided to cross-out
the initial working. Good thingas
it was incorrect! You should always
cross out what you do not wish to be
marked. Otherwise, the examiners will
mark what is written frst.
When ormulas are given in the
inormation booklet, marks are
usually not awarded or writing
these down without doing something
with them. In this question, correct
values had to be substituted into the
ormula or any marks to be awarded in
part (b).
The fnal answer is let unfnished,
so the student did not earn the fnal
mark or the answer o 30.
This answer achieved 5/5
(a) 3(0.9) 9 A1
(b) S =3_____1 0.9 r = 0.9 A1 A1
S =3
___0.1
A1
S = 30 A1
This is a model solution with clear
working shown and a fnished
answer. The correct value or the
common ratio, r, is written down,
and correct values are substituted into
the ormula or the sum o an infnite
geometric series.
8/3/2019 Maths SL Algebra
9/11
2. [Maximum mark: 6]
(a) Expand ( )x 2 and simplify your result. [3 marks]
(b) Find the term in x3
in ( ) ( )3 2 x x+ . [3 marks]
[Taken from paper 2 November 2008]
What are the key areas rom
the syllabus?
Binomialexpansion
Lawsofexponents
How do I approach the question?
(a) As the expansion is o a binomial nature, consider the binomial theorem.
You can use Pascals triangle or your GDC to fnd the binomial coefcients
quickly. Be especially attentive to any negative terms when expanding
binomials.
Texas Instruments Casio
(b) This question is linked to part (a) as its result is used to fnd the term in
x3. Although some may preer to multiply the 3xand the 4 over the entire
expansion rom (a), you only need to multiply the terms that makex3. It
is also important to remember that a term includes a variable, while acoefcient does not.
8/3/2019 Maths SL Algebra
10/11
0
5.Algebra
This answer achieved 1/6
(a) = 40 x420+ 41 x321+ 42 x24 + 43 x8 + 44 16 M1= x4+ (4 2 x3) + (6 4 x2) + (4 8 x) + (16)
= x4+ 8x3 + 24x2+ 32x + 16 A0
(b) (3x + 4)(x 4+ 8x3 + 4x2+ 32x + 16)
3x5+ 24x4+ 12x3 + 96x2+ 48x + 4x4+ 24x3 + 16x2+ 128x + 64
3x5+ 28x4+ 36x3 + 112x2+ 176x + 64 A0 A0
the term = 36 A0
This student uses the binomial
theorem to write a ull expansion o
fve terms. Although the expansion
is incorrect, having evidence o the
correct approach earns a mark.
The student uses a positive 2 in this
expansion where a 2 should be
used. In the correct expansion, the
second and ourth terms should be
negative as the 2 would be taken to an
odd power.
With an incorrect answer in part (a),
the student could still earn marks in
part (b) using the rules o ollow-
through. Unortunately, the 24x2ound in (a) is carelessly written as 4x2 in
(b), and so becomes 12x3 when multiplied
by 3x. Furthermore, 4 multiplied with 8x3
is incorrectly written as 24x3. Thus, no
ollow-through marks are earned.
8/3/2019 Maths SL Algebra
11/11
This answer achieved 3/6
(a) (x 2)(x 2) = x2 4x + 4 M1
x2 4x 4(x 2) = x3 4x2 4x 2x2+ 8x + 8 = x3 6x2+ 4x + 8
x3 6x2+ 4x + 8(x 2) = x4 6x3 + 4x2+ 8x 2x3 + 12x2 8x 16
= x4 8x3 + 16x2 16
= x4 (2x)3 + (4x)2 16 A0
(b) x 4 8x3 + 16x2 16(3x + 4)
= 3x5 24x4+ 48x3 48x + 4x4 32x3 + 64x2 64
= 3x5 20x4+ 16x3 + 64x2 48x 64 A1 (t) A1 (t)
= 16 A0
This answer achieved 6/6
(a) (x 2)4= x4+ 41 x3(2)1+ 42 x2(2)2+ 43 x1(2)3+ (2)4 M1= x4+ (8x3) + 24x2+ ( 32x) + 16
= x48x3 + 24x2 32x + 16 A2
(b) (3x + 4)(x 4 8x3 + 24x2 32x + 16)
72x3 32x3 = 40x3 A1 A1 A1
There is a recognizable attempt to
expand (x 2)4, which earns a mark
as this is a method that may lead to
a correct answer.
As the algebra o this method is long
and messy, errors are easily made
along the way. In this case, the
student miscopied thex2 4x+ 4 in
the frst line asx2 4x 4 in the second
line. Were this done correctly, ull marks
could have been earned. Still, as it is
time-consuming, using the binomial
theorem may be a more efcient
approach.
The terms 48x3 and 32x3 are
correctly ound rom the incorrect
answer given in (a). Marks are
awarded in ollow-through.
The question asks or the term inx3,
whereas this answer is given as its
coefcient, so the fnal mark is not
awarded. An answer o 16x3 would
be the correct term in ollow-through.
A completely correct expansion is
written that shows the pattern o
binomial coefcients and terms very
clearly. Brackets are used
eectively, which helps to keep the
algebraic simplifcations organized,
reducing the chance or error. A correct
and simplifed answer is given as
required.
As (x 2)4 is rom part (a), the
polynomialx4 8x3+ 24x2 32x+ 16
may be substituted into the
expression. Then the student
multiplies 3xby 24x2and 4 by 8x3 to fnd
the correct term inx3.
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