Arithmetic sequences

Arithmetic sequences

Season 02

Episode 08 Time frame 4 periods

Prerequisites :

Objectives :

•

Disover the onept of arithmeti sequene.

•

Find out the main formulae about arithmeti sequenes.

Materials :

•

Answersheet for the teamwork.

•

Lesson about arithmeti sequenes.

•

Exerises about arithmeti sequenes.

•

Terms from seven dierent sequenes.

•

Beamer

1 – Matching game 10 mins

Papers with numbers are handed out to the lass. Students mingle to nd the other

numbers thatouldbepart ofthe samesequene. Firsttermsare speiedwitha staron

the paper.

2 – Team work ^{45 mins}

Working inthe teamsfromthe previous part,students have tollananswer sheetabout

their sequene.

3 – Lesson 30 mins

The main results about arithmeti sequenes are shown with a beamer.

4 – Exercises Remaining time

Exerises about arithmetisequenes have tobe done ingroups of 3 or4 students.

Arithmetic sequences

Season 02

Episode 08

Document Answer sheet

1

^{. Write inthe ells belowthe rst ve terms of your sequene, inthe orret order.}

2

^{. Write inthe ells belowthe next veterms of your sequene.}

3

^{. Write in the ells below the terms of your sequene whose indies are given. For}

example, inthe rst ellyouhave towrite the

20

^{thterm in your sequene.}

20 25 50 100

4

^{. What isthe ommondierene}

d

^{between twoonseutiveterms inyoursequene?}

d =

5

^{. Let's note}

a 1

^{the rst term in your sequene,}

a 2

^{the seond term and so on. Give}

the notation not the value of the term next toeah of the terms below.

a 6 a 12 a 153 a ⁿ

6

^{. Find arelation between any term}

a n

^{and the next term.}

7

^{. Find arelation between aterm}

a n

^{, the ommondierene}

d

^{and the rst term}

a 1

^.

8

^{. Usetheformulayoufoundinthepreviousquestiontoomputediretlytheseterms.}

a 200 a 250 a 500 a 1000

9

^{. Find arelation between any term}

a n

^{and the term}

a 2

^.

10

^{. Find arelation between any term}

a ⁿ

^{and the term}

a 5

^.

11

^{. Find arelation between any two terms}

a n

^and

a m

^.

12

^{. Plae the rst ten terms of your sequene on the graph below. To do so, hoose a}

onvenient sale onthe

y

^-axis.

1 2 3 4 5 6 7 8 9 10

13

^{. What do you notie about the graph of this sequene?}

14

^{. Compute the sum of the rst ve terms of your sequene.}

15

^{. Find a relation between the sum of the rst ve terms, the number of terms, the}

rst term and the fthterm.

16

^{. Find a relation between the sum of the rst ten terms, the number of terms, the}

rst term and the tenth term.

17

^{. Find arelationbetween the sumof the rst}

n

^{terms, thenumberof terms,the rst}

term and the

n

^-thterm.

Arithmetic sequences

Season 02 Episode 08 Document Lesson

1 Denition and riterion

A sequene of numbers

(a ⁿ )

^{isarithmeti ifthe dierene between twoonseutive terms}

isa onstant number. Intuitively,togofromoneterm tothenext one, wealways addthe

same number.

Definition 1 Arithmetic sequence

A sequence of numbers ( a _n ) is arithmetic if, for any positive integer n , a _{n +1} − a _n = d where d is a fixed real number, called the common difference of the sequence. We can also write that

a _{n +1} = a _n + d.

This equality is called the recurrence relation of the sequence.

Proposition 1 Graph of an arithmetic sequence The graph of an arithmetic sequence is a straight line.

2 Relations between terms

Proposition 2 Explicit definition For any positive integer n ,

a _n = a ₁ + ( n − 1) × d.

This equality is called the explicit definition of the sequence.

Proof. First, this equality is true when

n = 1

^{, as}

a ₁ = a ₁ + 0 × d = a ₁ + (1 − 1) × d

. Then, suppose that it is true for a value

n = k

,meaning that

a _k = a ₁ + ( k − 1) × d

.Then, from the denition ofthe sequene,

a _{k +1} = a _k + d = a ₁ + ( k − 1) × d + d = a ₁ + k × d

.So theformula is true for

n = k + 1

^{too.Soit's truefor}

n = 0

^,

n = 1

^,

n = 2

^,

n = 3

^{,et, forall values of}

n

.

Proposition 3 Relation between two terms For any two positive integers n and m ,

a _n = a _m + ( n − m ) × d.

Proof. From the expliit denition of the sequene

( a _n )

^,

a _n = a ₁ + ( n − 1) × d

and

a _m =

a ₁ + ( m − 1) × d

,so

a _n − a _m = ( a ₁ + ( n − 1) × d ) − ( a ₁ + ( m − 1) × d ) = n × d− m × d = ( n − m ) d

. Therefore,

a _n = a _m + ( n − m ) × d

.

Theorem 1 Limit of an arithmetic sequence

The limit of an arithmetic sequence ( a _n ) of common difference d

• is equal to + ∞ when d > 0 ;

• is equal to −∞ when d < 0 ;

• is equal to a ₁ , trivially, if d = 0.

Proof.

•

Suppose that

a ₁ > 0

^and

d > 0

^{, and onsider any real number}

K

. Then, the inequation

a _n > K

, or

a ₁ + ( n − 1) d > K

,is solved by anypositive integer

n

suh that

n > ^{K − a 1}

d + 1

^.

Thismeans thatforanyrealnumber

K

,thereexist someinteger

N

suhthatfor any

n > N

,

a _N > K

.Thisisexatlythe denition ofthefatthat

lim a _n = + ∞

.

•

Suppose that

a ₁ > 0

^and

d < 0

^{, and onsider any real number}

K

. Then, the inequation

a n < K

, or

a ₁ + ( n − 1) d < K

,is solved by anypositive integer

n

suh that

n > ^{K −} _d ^{a 1} + 1

^.

Thismeans thatforanyrealnumber

K

,thereexist someinteger

N

suhthatfor any

n > N

,

a _N < K

.Thisisexatlythe denition ofthefatthat

lim a _n = −∞

.

•

The last situation,when

d = 0

^{,isobvious, asallterms areequal to}

a ₁

.

4 Sums of onseutive terms

Theorem 2 Sum of consecutive terms

Let ( a n ) be an arithmetic sequence, The sum S n of all the terms between a ₁ and a n , S _n = a ₁ + a ₂ + . . . + a _n

− 1 + a _n , or more precisely S = P ⁿ

i =1 a _i , is given by the formula S = n ×

a ₁ + a _n 2 .

Proof. The trik to prove this formulais to write the sumin two dierent ways, one based on

the rstterm, the other basedon the last term.Using the formula of proposition 1.2, we have,

on one hand

S _n = a ₁ + a ₂ + . . . + a _n

− 1 + a _n

S _n = a ₁ + a ₁ + d + . . . + a ₁ + ( n − 2) d + a ₁ + ( n − 1) d

and on theotherhand

S _n = a ₁ + a ₂ + . . . + a _n

− 1 + a _n

S _n = a _n − ( n − 1) d + a _n − ( n − 2) d + . . . + a _n − d + a _n .

Whenwe addthesetwoexpressionof

S _n

,alltermsinvolving

d

areanelledandweendupwith asmany timesthe sum

a ₁ + a _n

arethere wereof termsin

S _n

,so

2 S _n = n ( a _m + a _n ) S n = n ×

a ₁ + a _n

2 .

Arithmetic sequences

Season 02

Episode 08 Document Exercises

8.1

^{Asinglesquareismadefrom4mathstiks.Twosquaresinarowneeds7mathstiks}

and 3squares inarowneeds10 mathstiks.This proess denes asequene. Determine,

for this sequene,

1

^{. the rst term;}

2

^{. the ommon dierene;}

3

^{. the formulafor the generalterm;}

4

^{. how many mathstiks are in arow of 25squares.}

8.2

^{A triangular number is a natural number suh that the shape of an} equilateral

triangle an be formed by that number of points. For

n > 1

^{, let}

u n

^{be the dierene}

between the

n

^{-thtriangularnumber and the previous one.}

1

^{. Find the six rst triangular numbers.}

2

^{. Givethe 5rst terms of the sequene}

(u ⁿ )

^{. What kind of sequene ouldit be?}

3

^{. Write thesix rsttriangularnumbers usingonseutive termsof thesequene}

(u ⁿ )

^.

4

^{. Dedue a formulato nd diretly the}

n

^{-thtrianguler number.}

5

^{. Usethis formulatoomputethe}

36

^{thtriangularnumber. Isthis numberfamous for}

other reasons?

8.3

^{The third term of an arithmetisequene is -7and the 7}

^th

^{term is 9.Determine :}

1

^{. the rst term}

a 1

^{and the ommondierene}

d

^;

2

^{. the}

51 ^st

^term.

8.4

^{In an arithmeti sequene, the rst and seventh terms are}

x ²

^and

6 + x − 5x ²

^,

respetively.If the ommondierene is

x

^{, determine the possiblevalues of}

x

^.

8.5

^{Thetwelfthtermofanarithmetisequeneis5,andtheommondierenebetween}

suessive termsis 3. Determinewhihterm has avalue of 47.

8.6

^{In a given arithmeti sequene, the rst term is 2, the last term is 29and the sum}

8.7

^{A horizontal line intersets a piee of string at four points and divides it into ve}

parts, as shown below.

b b b b

1

2 3

4 5

Ifthepieeof stringisintersetedinthisway by19parallellines,eahofwhihintersets

it atfour points, nd the number of parts intowhihthe string willbe divided.

8.8

^{These two problems appear on the Rhind Papyrus. This papyrus was named after}

Alexander Henry Rhind, a Sottish antiquarian, who purhased the papyrus in 1858 in

Luxor, Egypt; it was apparently found during illegalexavations in or near the Rames-

seum (the memorial temple of Pharaoh Ramesses II). The British Museum, where the

papyrus is now kept, aquired it in 1864. It is one of the two well-known Mathematial

Papyri along withthe Mosow Mathematial Papyrus.

Problem 40. Divide 100 hekats of barley among 5 men so that the ommon dierene

is the same and so that the sum of the two smallest is 1/7 the sum of the three

largest.

Problem 64. Divide 10 hekats of barley among 10 men so that the ommon dierene

Season 02 • Episode 08 • Arithmetic sequences

⁷

Document 1

^{Cards for the mathing game}