Maths SL Algebra

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.


2/11

2

5.Algebra

Geometric sequences and series


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:

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n > 11.345

Trr, h minimum va lu n r which hinvmn ecd $10,000 i 12.


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.

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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:

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4/11

4

5.Algebra

The binomial theorem


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:

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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.


5/11

Exponents and logarithms


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.


6/11

6

5.Algebra

Infnite geometric series


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


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.


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.


8/11

8

5.Algebra


(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.


(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.


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.

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(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.


10/11

0

5.Algebra


(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.


11/11


(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


(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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Maths SL Algebra