SEQUENCE AND SERIES

Sequences

The Concept of Sequence

Explain he concept of sequence A Sequence is the arrangement of numbers or is alist of numbers following a clear pattern such that one number and the next are separated by comma (,).

Example: ay, 02, a5, 4

NB: Each number found in a Series or Sequence is called a fer.

Example 1

Find the next three terms in the following sequences. a 58111417, b.-3.7,6,10,9, e 12,47, a 2,9,20,35,

Solution

• You can see that each term is less to the next by 3

So next three terms are (17+3),(17#3+3) and 1743+3×3) Which are 20, 23, and 26 3 7 6 10 9 \ /\ /\ /\ / \ / \ / \ r% / \ f \ / \ \ / \ / \ f \ y \ /

V V V Vv

4 4 4 4 Altemately add 4 and subs tract 1.'The sequence then extends to 13, 12, 16 1 2 4 7 N eX /\ / \ / \ / \ / \ f \ fF \ f Kf \ / Ko \/ \ / \ /

V V V

We see thatthe difereneeis increasing by leach time, So the next three terms are I1, 16 and 22 2 9 20 35 f hi

LAK IX

\ f\ ff \ / \ / \ / \ / \ / / \ / \ A /

VV Vv v

T abl 15 The differences are inereased by 4 each time, so the nest three terms are 4, 77 and 104 a

Example?

Write dow the firs three terms in the sequences where the n erm is given by the formulae ans) ©

Solution

• 4n3

The Sequence is 1, 5, 9, we

• =

sua oat 3

The sequence is 1,3,5, Oo

Example3

The K* term of a series is 2+ 4 Find the sum of the frst four tems in the series

Solution

kel, Rede Ids

2, HB 3, He 3 N13 ke ded? d-20 So the series is $+8+13+20 and its sum is 46

Example 4

Find the n® term of the following sequences: (a)2,3,4,5.6,

Solution

So the nis n+1

• The numerators of fractions aro 1, 2, 3, 4 and the fractions

are 2, 3,4, 5 So the n™ term for numerator is n and that of denominator is n+ Combining them gives the n®*term of the fractions So the ni term is sea ter which is Then termis =>

• Write down the next three terms in the following sequences
• 1,5,9,13,17,
• 27, 24, 21,18

(€)5,8,9, 12, 43, ()1, 8,27, 64, aan OEP

• Find the first thre terms inthe sequence:

a Smt? b 13k e atintl a

• Find the sum of the first four terms ofthe series where the ke term is given by:

a Sk 6 RA

• Find the a tem ofthese sequences
• 2,4,5,6,
• 4.5,6,7,

(©) 10, 20, 30, 40,

• 2,4,6,8.
• 13, 24,3545,

2h35,4i55, An Arithmetic Progression (AP) and Geometric Progression (GP) Identify an arithmetic progression (AP) and geometrie progression (GP) When the series or sequence is such that between two consecutive terms there is a difference which is fixed, then the series or sequenee is called an arithmetic progression (A.P) The fixed difference (number) between two consecutive terms is called the common difference

Example S

In the sequence 4, 7, 19, 13, 16 there is a common difference which is

Note that in arithmetic progression (A.P) the difference between two successive terms is always the same, Sometimes numbers may be decreasing instead of increasing, the arithmetic sequence or series while terms decrease have a negative number as a common difference

Example 6

‘The common difference of the sequence 6, 4, 0, -2 is

So the common difference is -2. In general if Ay, As, As. Aa, ‘An are the terms of the arithmetic sequence , then the common difference is

=A ALAS AS Ae AGE. r

Example 7

For each of the following sequences, find the common difference and write the next two terms,

• -2,5, 12, 19, 26,

)45,35,23,

Solution

So the next two terms are 33 and 40,

The common difference is —2, 3 (8 3, -5)4(-5 The next two t G =)) ¢ =))+()) ye next two terms are: (* + (—*))and (7 + (4S) Which are— and — 2 a

• Find the common difference far each ofthe following sequence

a 11, 14,17,20, b. 2,4,6,8,10, © 04,011,011, 01111, GL ysyAR yr yoo, yt 2 State whetherthe following sequence are arithmetic or not a 2,58, 1114, b1,3,4.6.7,9.10, &y.ytx.yt2e,yt3n, 3, The temperatureat a mid day is 3°, and it falls by 2% cach hour. Find the temperature at the end of the next four hours,

Geometric Progression (GP)

When the series or Sequence is such that between two consecutive terms there is ration which is fixed, then the series or sequence is called a geometric progression (GP) The fixed ratio(number) between two successive terms is called the common ratio (F)

Example 8

In 2, 4,8, 16,32, ‘There is a common ration which is AeisteBs 248 a6

Note that like in arithmetic progression (A.P), in geometric progression (GP) the common ratio does not change. Also the terms may be decreasing instead of increasing, the geometric sequence or series whose terms decrease have a positive common ratio which 1s Jess than 1 for the progression with positive terms,

So the common ratio is 5. Generally if Gr, Gz, Gs, Gs,…… Gaare the terms geometric sequence then the common ratio is

ae Se Ge

TG a2 a3 ant

Example 9

For cach of the following sequence find the common ratio.

• 3,6,12, 24

betel (15 Rpe pe

Solution;

• 3,612, 24,

G1 G2” Gs

ise sites

m 4,32,3 om peSe%i% Lalor (tat)or (242 rt The common ratio (t) = ~

Example 10

terms (2)2,3,44, 62

• 10, -6, 25, -1.25

Solution

• Since 2,3, 6 …. is a geometric sequence, it has a common ratio (f) which is

found as follows aS2G_L4 a” a2 a3

m3

The next term is found by multiplying the term considered to be the last term by the common ratio a1

oo 6 © ond 282

(©) 10,5, 25,-.25. 5 25 125 petetees

The common ratio (t) => The next two terms are: (-1.25 )) and (-1.25 x 9) x (S) 5 ans

gas

• Which of the following sequences are geometric

a 12,4816 b —-2,6,18, 54, 162, © LMA, dS ,2x', ad, 80 e 1,2,4,7,10, £ 01,02,03,04.05, & 36,9, 1215,

• Find thecommon difference for each ofthe following geometric progressions (G-P)
• 1, -0.5, 0.25, -0.125,

{b) 4x, 4×7, 4×3, 4×4, – ott 24 2'4"e' 16"

• 24,22, 44,88, ssanse
• Pind thenext term of the sequence 2, 10, $0, 500.

4, The populationof a town is decreasing so that every year the population declines by a quarter If the population is originally 100,000. What will it be after 5 years? ‘The General Term of an AP Find the general term of an AP IAL, An An ‘Agate the terms of an arithmetic sequence, then there is a ‘common difference d which is given by

d= A2—Ar=A3—A2= Ae Aa

d= Ase or d= As—Asor d= An—Ans

Means A2= At d

As= Aad

Bul Ao=Ared

But. As—Ar + 2d which means

Ag=[ Art2d]+d

Avt3d Putting into consideration this pattern, itis true that

Ast Art dd

Aa Ai + Sd

Where Apis the n® term The n term of the sequence with first term A; and common difference d is given by Find theformula for the n"* term of the sequence 8 . 9.5, 11, 12.5, 14. 155,

Solution

Or Ac-As

Note that the n' term gives every term in the sequence,

Example 12

The 5* term of an arithmetic sequence is 11, and the §" term is 26, Find the first five terms,

Solution

Given that

From AaAtt (nl) d

So Avs4d=11 a)

Solving for At and d fimuttanoously gives

Aas -9+ (1-1) x5

AnSe14 A514 Son=t

The first five terms are -9,-4, 1,6, and 11

Example 13

The 8* term of an arithmetic sequence is 9 greater than the 5" term, and the 10" term is 10 times the 2” term. Find a. The common difference (4) b 20 term.

Solution

Let As, Az. As, An

Be the terms of the given sequence As > AS by 9 means

‘Aiais 10 times the second term means

AsAread

So Ars7d-(Artdd) =9

‘Also, Auo=10Azmeans Ar/9d=10(Ar+d)

-a Az

ard Att Therefore the common difference (4) is 3 To get the 20" term, use the n™ term Lo. AgsAv+ (n-1) d

The 20 term is 22

The General Term of GP

Find the general term of GP HG, G:, Gs .G, are the terms of a geometric sequence, then they have a common ratio (F) which is given by

Ge Sen Me

So Gs= (Gur) r

Where Gir =Ge

G=Gir Ge (Gi) r

Where Gir?=Go

GeGit

We have soon that GrGi=Gue G=Gu'=Gir GeGie Geir GeGirt GeGis GerGir Where Gris the n® term,

Example 14

Find the formula For the n' term of each of the following geometric sequence a 26,18, 54 b 4,2, 1,-0,0.25 a

Solution

• 2,6, 18, 54
• From the sequence 2.6, 18,54
• Gr=2,62-6 6:18 and Ge-54

Gi G2

6_ 18_ 54

26 ie

But Go =Gurrt

Ge-2x (38) xt Gaz 2 x 3°

(©) 4,-2,1, -05, 0.25, or4

Se Bes IL, a” a2 2 re

but Ge= Gum

mee (=)

Example 15

Considering that, Find the 136 term of this sequence

Solution;

ta Given the sequence3, 1, 4, 3, G3 Get Get

Ge 3, Gu?

¢ a

Sa Setnd

ai G23 4 rt from Ga=Girt

So Grs=3(2)!™1

G-3(2)" = (=a The 136 termis (2)! or 34"

• Inthe arithmetic sequence, the 17 term is 30 and 9*erm is 42 find the first three terms.

2, In the Avithmeticsequence the third term 12 and the 9*erm 24. Find the o! term of the sequence and use ito find the 15" term. 3, Find the1S" term of the sequence 5, 10, 20, 40

• population isincreasing and every year itis multiplied by 1.03, IFit stats off at 10,000,000,

What will itbe after n years? ua

• The first termof the geometric sequence is 7 and the common ratio is 4. What is the 9 term of

this sequence? Series The Formula for a Sum of an Arithmetic Progression Derive the formula for a sum of an arithmetic progression When the terms are separated by addition (+) sign, there we have what we call a series.

Example: 2446-84,

Isa series with the first term (A1) 2 and common difference (d) 2 Ivis possible to establish a formula for the sum of the fist n terms of the arithmetic progression. Let S, denote the sum of the first m terms of the arithmetic series Consider the sum of the first 5, terms of arithmetic progression (AP) whose first term is 1 and whose common difference (4) is 1

So Ss= ArtAatAstActAs

Si2H HS 0)

The first case is the sum of five terms which are increasing from up to 5 while the second case shows the same sum but the terms are decreasing from 5 to 1 If you add (1) and (2) together, you find that

25.— 616164646 Dividing by 2 each side gives as

2142434445515 Now for any number of terms (n) SnmArt+Aa+As+. +A nitAn Which is the same as

So=An + AnstAnat… AZtAt

Also we can write

Which means

«The sum of n terms in arithmetic progression is

Example 16

Findithe sum of the first 20 terms ofthe series

Solution

From the series above

Example 17

Find the sum of the series 4¢7710+13+ +304

Solution

To use the formula for summation of n terms, you must know how many terms are there, Le finding the value of n; Now

AsAc+ (Dd

304-341

‘Thorofore we are required to find the sum of the 101 torms of the given series.

= + Gos)

The sum of 4+7+10+13+ +304 is 16,554,

Example 18

How many terms of the series 1+3+5+7+ are needed to make the sum of 1697

Solution

Art a2 n?

2-13 terms are required

• Find thesum of the first 20 terms of the series

a Sef b. 19+t613+10-7~

• Find thenumber of terms and the sum ofthe series

a ses b. 4037340314 257 3, The Sumof the first 10 terms of an arithmetic progression (A.P) is 40, and the sum of the next 10 terms is 80. Find the sum of the fitst five terms of the series.

• One dayFrola spends 40 minutes of her home work. The length of time she spends inerease by

4 minutes each day, Find the total length of time she spends after eight days.

‘The Arithmetic Mean

Calculate the arithmetic mean Remember thatthe arithmetic mean (M) of n numbers is found by adding them and then dividing the sum by n, ¢g the arithmetic mean of abc and d is atbtcta veo The Formula for the Sum of a Geometric Progression Derive the formula for the sum of a geometric progression Geometric series are the series that can be written as

Gi+GaGst G

Example: 264+8+16+ Ge

Or 153+9427481+ Suppose we want to find the sum of 1+3+9+9+27+81+ Soolt349427481 a It we multiply s, by the common ratio(r), we have.

Subtracting (1) trom (2) aves

so= Bata In general, for a series with terms Gr+G2+Gat Gn and common ratio("#1),

So= Gi+GireGut+ Gu

Sax Cart Girt Gir +r

Now 1Se- S1= Gu?- Gr

Sa- Sa= Gr (4)

vt fe ra NB; If-1<r <1, itis easier to use the formula in the form of uo 010-1") ra

Example 19

1, Find the sum ofthe geometric series 2-4+8+ +2048 ma

Solution

G2, 2

Sn?

From Gr=Gu"

2ns2e nit

Also Sx =o

The sum is 4094

Example 20

Find the som of the frst 8 terms of the series 5+20+80+320+ a2

Solution

as Se?

seox gq Sx(E8S36-1) _5x65,536

-The sum is 409,226

• For each ofthe following series, find the number oF terms and hence the sur of the series

b L2H8: +1024

• Find the sum of the first 10 terms of the series
• 4s2410% +

(o)4.2et te

• Masania sots off on a long Journey. The first day ho walks 30km, but the distance

he walks each day is 10% less than on the previous day. Find the tolal distance he has walked after 12 days

The Geometric Mean

Cateulate the geometric mean ‘The Geometric mean (GM) of n positive numbers is found by taking the a root of their product,

Example 21

‘The Geometric mean ofa, b, ¢ and d is GM= sJabed Therefore the arithmetic mean of 3 and 5 is Mom ang

But the geometric mean of 3 and 5 is

GM V3x5

GM=VI5=3.87

The arithmetic mean and geometric mean can be used to check that a sequence isan arithmetic ‘or geometric respectively

FACTS

1 Ifa, band € are consecutive three term of arithmetic progression (A.P), then b is the arithmetic mean of and ¢

• Ifa, b and ¢ ate three consecutive terms of geometric progression (G.P), then b is the

geomeitic mean (GM) ofa and e Proof: 1s

• Suppose a, b and c are the three consecutive terms of an arithmetic

progression then

So ba-cd babrarc

p= as required

• Lot a, b and c be the consecutive terms of geometnc progression (G.P)

sob=£ eb brb=axc beraxe b=Vaxe as required

Example 22

Find the arithmetic and geometric means of ws (@) Sand 42 (b)5.8,14

Solution

(@) Arithmetic mean wettest ays

METS

<The arithmetic and Geometric means of 3 and 12 ate 7.5 and 6 respectively.

• m= 8* ses

The arithmetic and geometric mean of 5, 8 and 14 are 9 and 8.24 respectively.

• Find the arithmetic and geometric means of the following,

bax. Ix © 4a,25a

• The arithmeticmean and geometric mean of two numbers are 7.$ and 6 respectively. Find the

two numbers

Compound Interest

‘Compound Interest using Formula Cateutate compound interest using formula Suppose money is invested or borrowed. At the end of a year, interest is calculated. Suppose this interest is added to the original principal, and at the end of the next year interest is added to the new principal. This process may be continued for a number of years.

This process is called COMPOUND INTEREST. When money is invested at a compound interest, he amount of money increase as a geometric sequence

Example 23

Solution

So after one year the amount is 20,000 x 1.06 After two years: 20,000×1.06×1.06

After 3 years: 20,000x (1.06) Alter 4 years. 20,000x (1.08)* After 5 years: 20,000x (1.08) 226,765.

Now let the principal be P, the rate RY% and the time in years be a a7 PR The interest im one year is The amount after one year is

PR R

100 100 R

Which is P(1 +4)? and so on,

Example 24

does she have at the end of the 10" year?

Solution

She has made 10 different investments each giving different amount of interest. The 1 investment has hae 10 years oF interest, hence it is 10,000 x (1 05)" The 2™ investment has had 9years of interest. So it is 10,000 x (1.05)° The 3" investment has had 8 years of interest, Hence itis 10,000 x (1.05)" Following this pattem, The 10" investment has had 1 year of interest. Hence it is 10,000.05.

‘The sum ofall these amounis is given by. 10,000×1.05" 10,000 x1.057» 10,000×1.08" +10,000×1.05 This geomet seties with first term 10,000 x 1.05 andl comm ratio 1.05, Hence the sum is 2200282 _ 139 068

1, Find the total amount of the following savings if they eam compound interest

• 100,000/= for 2 years a 6% pa

• A population is increasing a 2% it stats at 10,000,000 what wilt be after 20 years
• At the beginning of each year 600,000/= is invested at 6% compound interest Find the total

value ofthe investment at the end ofthe 15* year. ws