region iv a module 1 sequences revised 2

Regional Mass Training of Grade 10 Regional Mass Training of Grade 10

Teachers of The K To 12 Teachers of The K To 12

Enhanced Basic Education ProgramEnhanced Basic Education Program

Venue: De La Salle University-Dasmariñas City of DasmariñasDate: May 3-8, 2015 (First Batch) May 10-15, 2015 (Second Batch)

Module 1Sequences

May 12, 2015

Objectives of the Session

DepED – Region IVA CALABARZON

At the end of the session, the teacher-participants are expected to:

1. demonstrate knowledge of key concepts on sequences;

2. appreciate sequences as functions; and3. apply sequences in solving problems.

Priming


Guess the next term based on the patterns!!!

What comes next?

O T T F F S S E N _?

T for tenT for ten

DEPARTMENT OF EDUCATION

A E I M Q _?


U

J M M J A O _?


D - DECEMBER

T Q P H H O N _?


D - DECAGON

1, 4, 9, 25, - - -


36

2 4 5 7 9 11 _?


13 – LOCAL TV CHANNELS

R O Y G B I _?


V – VIOLET

2 3 5 7 9 11 13 17 _?


19

1 9 25 49 81 _?


121

A B C E F G _?


I

P E M D A _?


S

2 7 12 17 22 _?


27

2 3 5 6 8 9 _?


11

A J S _ ?


N – NOVEMBER, MONTHS WITH EXACTLY 30 DAYS

A, 1, B, 2, C, 3, - - - -

D


0, 1, 16, 81, - - - -

625


Congratulations!


Group Activity


On a manila paper, give relevant information on sequences presented based on the slides. Give some properties related to the sequences. After 5 minutes, the group output will be presented to the class by the assigned presentor.

Analysis


•1. What have you been thinking/feeling after you finished the activities?

•2. If you will place yourself in one of your students’ shoes, what will they feel after doing the activities?

Lesson 1: Arithmetic Sequence


In this lessons, the learner will learn to:-generate and describe patterns-find the next few terms of a sequence-find the general or nth term of a sequence-find the arithmetic means between terms of an arithmetic sequence-determine the sum of the first n terms of a given arithmetic sequence-solve problems involving arithmetic sequence

Lesson 2: Geometric and othe Sequences


In this lessons, the learner will learn to:

- illustrate a geometric sequence-differentiate a geometric sequence from an arithmetic sequence-determine the nth term of a given geomtric sequence-find the geometric means between terms of a geometric sequence-illustrate other types of sequences like harmonic sequence and Fibonacci sequence-solve problems involving geometric sequence

Sequences

DepED – Region IV-A CALABARZON

SEQUENCES

NUMBER SEQUENCES

NON-NUMBER SEQUENCES

(OTHER TYPES)

ARITHMETIC SEQUENCES

GEOMETRIC SEQUENCES

REAL-LIFE PROBLEMS ON SEQUENCES

NTH TERM OF A SEQUENCE

SUM OF THE N TERMS OF SEQUENCES/

INFINITE GEOMETRIC SEQUENCE

ARITHMETIC/ GEOMETRIC MEANS

NEXT TERMS OF A SEQUENCE

What is a sequence?


Definition: A sequence is a function whose domain is the finite set {1, 2, 3, …, n} or the infinite set {1, 2, 3, …}

Recall: A function is a correspondence from a set A to a set B which matches a unique member of set A to a member of set B.

It can be a pattern of :

• objects• music• letters• or numbers

A sequence is a pattern.

What is a sequence?

Terms of Sequences


A sequence is the “correspondence” or “rule” itself. For number sequences, the formula may be given as an equation.

Terms of Sequences


CONVENTION: The terms of the sequence is loosely referred to as the sequence itself.

( ) nnf 3=

Ex: From the set of natural numbers n match 3n

Terms of Sequences


( ) nnf 3=

},,{: 321fofDomain

},,{: 963fofRange

We refer to {3, 6, 9} as the sequence for convenience onlyconvenience only although 3, 6, and 9 are the terms in the range of the sequence. The sequence is the correspondence itself.

SPECIAL TYPES OF NUMBER SEQUENCES


1. Arithmetic Sequence

2. Harmonic Sequence3. Geometric Sequence4. Fibonacci Sequence

ARITHMETIC SEQUENCE...


An arithmetic sequence is a sequence whose every term, except the first, is obtained by adding to the previous term a constant called the common difference.

Arithmetic Sequence


Notations:

a1= first term of the sequencean = nth term of the sequenced = common differenceAn = sum of the first n terms of the arithmetic sequence

Arithmetic Sequence


a2 = a1 + da3 = a2 + d = a1 + 2d

an = a1 + (n-1)d

.

.

.

sequencetheoftermsta 1=1

sequencetheoftermnthan =

d = common difference

An = sum of the first n terms of the sequence

Arithmetic Sequence


Let An = a1 + a2 + · · · + an

An = an + an-1 + · · · + a1

Adding the two equations, 2An = a1 + an + a2 + an-1 · · · + an+ a1

2An = (a1 + an)+ (a2 + an-1)· · · + (an+ a1)2An= n(a1+an) (why?)

)( naan

nA +12=

ARITHMETIC SEQ.


Now take note that an = a1 + (n-1)dHence

An = (n/2)[2a1+ (n-1)d]

This is another formula for the sum of the terms in the arithmetic sequence

ARITHMETIC SEQ.


Example:1.What is the nth term of the arithmetic sequence 7,9, 11, 13, 15, 17, . . . ?

Answer : 2n+5

2. If three arithmetic means are inserted between 11 and 39, find the second arithmetic mean.

Answer : a2 = 25.

GEOMETRIC SEQUENCE...


A geometric sequence is a sequence whose every term, except the first, is obtained by multiplying a constant called the common ratio.

Example : 32, 16, 8, 4, 2, 1

Geometric Sequence


Notations:a1= first term of the sequencean = nth term of the sequence = a1rn-1

r = common ratioGn = sum of the first n terms of the geometric sequence

Geometric Sequence


Multiplying both sides by r,

Subtracting this from the previous equation,

where r≠1

Gn= a1 + a1r + a1r2 + • • • + a1rn-1

rGn= a1r + a1r2 + a1r3 + • • • + a1rn-1 + a1rn

(1-r)Gn= a1 - a1rn

Geometric Sequence


where r≠1

The sum of a geometric sequence is

What if r =1 ?If r =1 , Gn = a1 + a1 + a1 + · · · + a1

Gn = na1

What if r =-1 ? (Hint: Consider the case when n is even and when n is odd.)

Geometric Sequence


Examples:

1. What is the 6th term of the geometric sequence ?

Note that r = 5, n = 6 , an = a1rn-1

( )250

5252

6

56

516

=

=

=

a

a

raa The 6th term of the geometric sequence is 250.

Geometric Sequence


2. Find the sum of the geometric sequence where the first term is 3, the last term is 46875 and the common ratio is 5.

Answer : 58,593 (Why?!)

Infinite Geometric Sequence


Now, what is the sum of an infinite geometric sequence ? a1 , a2 , a3 , . . . a1 , a1r , a1r2 , . . .Originally we have

ra

G

,rif

n −=

<<−

1

11

1

r ≠ 1

But as n approaches infinity,

Geometric Sequence


Example : What is the sum of the geometric sequence

+−+−91

31

13

Solution :

31

31−== r,a

49

31

1

3 =

−−

=nG

?

HARMONIC SEQUENCE...


A harmonic sequence is a sequence whose reciprocal of every term forms an arithmetic sequence

Example : ,...81

,61

,41

,21

FIBONACCI SEQUENCE...


A Fibonacci sequence is a sequence whose every term, except the first two terms, is obtained by adding the two previous terms. The first two terms may be 1, 1 or 0, 1.

ACTIVITY


What to look forwhen looking for the rule

defining a sequence?

Here is a sequence of objects

Correct! It would be a mouse.

What animal would come next in this sequence?


Here is a sequence of sounds

What sound will come next in the sequence?

Correct! It will be the cow mooing.

Hear me!! (click each one)


Here is a sequence of letters.

What letter will come next in the sequence? Look carefully at

the letters.

a a b c a a b c a a b c a

Correct! It‘s the letter a.


Here is a sequence of numbers.

What number is going to come next in this

sequence?

0 2 4 6 8 10 12 14 16

Correct it is the number

18DEPARTMENT OF EDUCATION

•Decide what number should come next

Tip! To find the next number in a number sequence:

•Put your number into the sequence and check that it looks correct

•Find the pattern


Play

and finish these number

sequences.

Look at the number sequences and see if you can

work out the missing number.


Find the number!

1 3 5 7 9 13 15 17

It is the number 11.

11


Find the number!

Correct! It is the number 15.

0 5 10 ____ 20 25 30 35 4015


Find the number!


17 15 13 ____ 9 7 5 3 11


Find the number!


25 35 45 ___ 65 75 55


34 44 54 64 ___ 84 94

Now race on to the last number

sequence.

Find the number!

Correct! The missing number is 74.

74


42 47 ___ 57 62 ___ 72 ___ 8252 67 77

The answers are: 52 67 and 77


ACTIVITY!!


Answer the following : p. 18 Activity 10 Groups 1- 3 : odd numbers Groups 4 & 5 : even numbers

p. 40 Activity 9

ACTIVITY!! Puzzle Sequences!!


On a whole sheet of Manila paper, your group is tasked to create a puzzle for the enjoyment of kids and adults alike!!

The puzzle may include terminologies on sequences and may involve solving problems on sequences.

Suggestions:

CROSSWORD PUZZLESNAKES AND LADDERS

NOTE: Please include the mechanics of the puzzle.

Closure

https://www.youtube.com/watch?v=SjSHVDfXHQ4


Watch the Tedtalk on youtube and write a brief reflection on the talk. Include how you could relate sequences to your students using localized or contextualized examples!

https://www.youtube.com/watch?v=SjSHVDfXHQ4

Closure


Instruct a wise man and he will be wiser still; teach a righteous man and he will add to his learning. Proverbs 9:9

Acknowledgement of Sources:

www.worldofteaching.comwww.youtube.comwww.xpowerpoint.com


http://www.worldofteaching.com/

http://www.youtube.com/

http://www.xpowerpoint.com/

http://www.myniceprofile.com/thank-you-149669.htmlDepED – Region IVA CALABARZON