number sequences and series
TRANSCRIPT
BY : ARITHMETIC GROUP
Number Sequences and Series
(Barisan dan Deret Bilangan)
Arithmetic Group1. Crisdina Suseno
2. Ihsan Prasetya
3. Ramadhina Putri Irza
4. Tanisa Pradani Resna
Standard competence Basic Competence
To comprehend the sequence and series
number, and its application to solve
a problem.
To determine the simple number sequence.
To determine the n-term of arithmetic and geometric sequence.
To determine the sum of n of the first term of geometric and arithmetic series.
To solve problems related to the sequences and series.
Number Sequence and Series
In this chapter we’ll learn…
Number Sequence
Number Series
Pattern of number
Introducing
Before we learn about number sequence and series, try to guess the next three number
from the problems.
1. 3, 6, 9, 12, 15, 18,...2. 25, 19, 13, 7, 1, -5, …3. 3a, 5a, 7a, 9a, …4. Try to explain and give
example of real number !5. If the first number is 2,
and the next number is….
! SIMPLE TEST !
POLA BILANGAN
Pattern of Numbers
Definition of Pattern of Numbers
The pattern of numbers is defined as the orderly arrangement of numbers.
OrSusunan bilangan yang memiliki keteraturan.
Pattern of Odd Numbers
Pola Bilangan Ganjil
1, 3, 5, 7,...
Susunan batang lidi menunjukkan pola bilangan ganjil
1
75
3
3 1Pattern of Odd Numbers
The pattern of odd number is 1, 3, 5, 7, 9, …
Formula Of Pattern Of Odd Numbers
The formula of an odd numbers pattern for the n-order is….
2n-1
Formula
The sum of n of the first odd numbers is n² which is written……
1 + 3 + 5 +….+ (2n-1)
Where n is natural number.
Example
1. Find the sum of 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19.
Answer :There are 10 terms.
n²So,
10² = 100
exAmpLe
Example
2. Find the sum of 9 + 11 + 13 + 15 + 17 + 19.Answer :
From the first question we know that sum of the first 10 terms is 100. Now we lost the first 4 numbers of the pattern,
that are 1 + 3 + 5 + 7 = 4² = 16 so,9 + 11 + 13 + 15 + 17 + 19 = (1 + 3 + 5 + 7 + 9
+ 11 + 13 + 15 + 17 + 19 ) – ( 1 + 3 + 5 + 7)
= 100 – 16= 84 exAmpL
e
Example
3. Calculate the sum of the second 15 terms of odd pattern numbers.
Hitunglah jumlah dari 15 bilangan ganjil kedua. Answer : The sum of the first 30 terms - the sum of the first
15 terms= 30² - 15²
= 900 - 225= 675
exAmpLe
Pattern of Even Numbers
Pola Bilangan Genap
2, 4, 6, 8,...
Susunan noktah menunjukkan pola bilangan genap
2 864
4 2Pattern of Even Numbers
The pattern of even number is 2, 4, 6, 8, 10, …
Formula Of Pattern Of Even Numbers
The formula of an even numbers pattern for the n-order is….
2n
Formula
The sum of n of the first even numbers is
n(n+1) which is written……
2 + 4 + 6 +….+ 2n = n(n+1)
Where n is natural number.
Example
Find 2 + 4 +6 + 8 +…
50 termsAnswer : = n(n+1)= 50(50+1)= 50(51)= 2550
exAmpLe
Example
In the sequence 2, 4, 6, …100 how many the terms?
Answer :100 : 2 = 50 terms
exAmpLe
Pattern of Triangle Numbers
Pola Bilangan Segitiga
1, 3, 6, 10,...
The Figures of The Pattern of Triangle Numbers
1 10
3 6
Pattern of Triangle Numbers
The pattern of triangle number is 1, 3, 6, 10, …
Formula Of Pattern Of Triangle Numbers
The formula of an triangle numbers pattern for the n-order is….
n(n+1)2
Example
1. Find the 15th number of the pattern of triangle numbers.Answer := n(n+1)= 15(15+1)= 240
exAmpLe
Pattern of Square Numbers
Pola Bilanga
n Persegi
1, 4, 9, 16,...
The Figures of The Pattern of Square Numbers
1 16
4 9
Pattern of Square Numbers
The pattern of square number is 1, 4, 9, 16, …
Formula Of Pattern Of Square Numbers
The formula of an square numbers pattern for the n-order is….
n²
Example
1. Determine the 25th term in pattern of square number.
= n²= 25²= 625
exAmpLe
Answer
3. Determine the 30th term from the pattern of square number.Answer := n²= 30²= 900
exAmpLe
Pattern of Rectangle Numbers
Pola Bilanga
n Persegi Panjang
2, 6, 12, 20,...
The Figures of The Pattern of Rectangle Numbers
2 12
6
Pattern of Rectangle Numbers
The pattern of rectangle number is 2, 6, 12, …
Formula Of Pattern Of Rectangle Numbers
The formula of an rectangle numbers pattern for the n-order is….
n(n+1)
Example
1. Determine the 21th term from pattern of rectangle number. Answer := n(n+1)= 21 (21+1)= 21 (22)= 462
exAmpLe
Pattern of Pascal Triangle Numbers
Pola Bilanga
n Segitiga Pascal
Pascal Triangle
To sum the numbers of Pascal triangle we’ll use the formula :
2n-1
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Example
1. Find the sum of the following Pascal triangle numbers lines on:a. the 5th lineb. the 8th line
Answer :a. n = 5, so 2n-1
= 25-1
= 24
= 16exAmpLe
Answer
b. because n = 8, so 2n-1 = 28-1
= 27
= 1282. Find the lines of the pattern of Pascal triangle numbers if
the sum of the lines is 64.Answer : 64 = 2n-1 26 = 2n-1 6 = n - 1 n = 6 + 1 n = 7
exAmpLe
Example
3. Factorize (x + y)4 and what is the 2nd and 4th coefficients ? Tentukan hasil dari (x + y)4, kemudian tentukan pula koefisien suku ke-3 dan suku ke-7 ?Answer :(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4
From algebraic form above we know that the 2nd coefficient is 4 and the 4th coefficient is 4.
exAmpLe
Exercise
1. The 14th terms from the pattern number 1, 3, 6, 10, … is….Suku ke 14 dari barisan bilangan 1, 3, 6, 10, … adalah…
2. The pattern numbers in the sequence 2, 6, 12, 20, 30, …, is…Pola bilangan pada barisan bilangan 2, 6, 12, 20, 30, …, adalah…
3. The next 4 terms from 1, 3, 6, 10, ...is…Empat suku berikutnya daribarisan 1, 3, 6, 10,… adalah…
4. In the pattern of Pascal triangle numbers, the sum of the numbers on lines 10th is…Pada susunan bilangan-bilangan segitiga pascal, jumlah bilangan yang terdapat padabaris ke-10 adalah….
Exercise
5. The n-term from the pattern 2, 6, 12, 20, 30, …is… Suku ke-n dari barisan 2, 6, 12, 20, 30, …adalah…
6. In a meeting room there are 14 chair at first line, 16 chair at the second line, 18 chair at third, and hereinafter increase 2 chair. If the room has 25 chair line. How many chair entirely ?
7. Find the next five terms from the pattern of 9, 16, 25, 36, 49, ….Tuliskan 5 suku berikutnya dari pola bilangan 9, 16, 25, 36, 49, ….
8. Find the next three terms of the pattern of 6, 10, 15, 21, 28, …Tentukan 3 suku berikutnya dari pola 6, 10, 15, 21, 28, …
Exercise
9. Find the sum of the 11th lines Pascal triangle !
10. Factorize the algebraic forms by using Pascal triangle. What is the 3nd and 4th coefficients of (x + y)6
BARISAN BILANGAN ARITMATIKA
Arithmetic Sequence Number
ARITHMETIC SEQUENCES
An arithmetic sequence (or counting sequence) is defined as a sequence which is achieved by adding or subtracting previous terms with constant number.
Characteristic of arithmetic sequences : Is regular sequence number Have the same divergent No plus (+) or minus (-) sign in sequences
Example :
2 4 6 8 10 12
+2 +2 +2 +2 +2
1st term
2nd term
3rd term
4th term
5th term
6th term
Math Difficult Terms
HOW TO FIND THE n-term
a, (a+b), (a+2b), (a+3b), (a+4b), …
Un = a + (n-1) b
HOW TO FIND THE DIVERGENT
b = U2 - U1
HOW TO FIND THE MIDDLE TERM
Ut = 1/2(a+Un)
If b > 0 it means that the arithmetic sequence is upIf b < 0 it means that the arithmetic sequence is down
EXAMPLE :
Given : Arithmetic sequence : 3,5,7,9,11,13,…Ask : Divergent and the 10th term
Answer :
divergentb = U2 – U1
= 5 – 3= 2
The 10th termUn = a + (n-1) bU10 = 3 + (10-1)2U10 = 3 + 18U10 = 21
exAmpLe
EXAMPLE :
Given : The first term of arithmetic sequence is 8, andthe divergent is 5
Ask : the 15th term, and the middle term if the total of term is 15
Answer :
The 15th term : The middle term:
Un = a + (n-1) b Ut = 1/2(a+Un)U15 = 8 + (15-1)5 = 1/2(8+78)
= 8 + (14)5 = 1/2(86) = 8 + 70 = 43= 78 exAmpL
e
EXAMPLE :
Given : 2/5, 5/7, 8/9, 1, …Ask : the n-termAnswer :
2/5, 5/7, 8/9, 1,… = 2/5, 5/7, 8/9, 11/11
Numerator = 2,5,8,11 denominator = 5,7,9,11
b = U2-U1 b = U2-U1= 5-2 = 7-5= 3 = 2
Un = a + (n-1)b Un= a+(n-1)b= 2 + ( n-1)3 = 5+(n-1)2= 2 + 3n-3 = 5+2n-2= 3n +2-3 = 2n+5-2= 3n -1 …….......(i) = 2n + 3 …………(ii)
So, the n- term is = i = 3n-1 ii 2n+3 exAmpL
e
Check your understanding
!!
1. The 40 term from 7,5,3,1,… is …
solution :
b = U2-U1= 5-7= -2
Un = a+(n-1)bU40 = 7 + (40-1)-2
= 7 + (39)-2= 7 + (-78)= -71
Check your understandin
g !!
2. Pada sebuah gedung pertunjukan, banyak kursi pada baris paling depan adalah 15 buah, banyak kursi pada baris di belakangnya selalu lebih 3 buah dari baris di depannya. Berapa banyak kursi pada baris ke-12 ?
Solution :15, (15+3), (15+6), … , U12
a= 15 b=3 Un = a+(Un-1)bU12= 15 +(12-1)3
= 15 +(11)3= 15 + 33= 48
Check your understandin
g !!
3. Arithmetic sequence is 2,5,8,14,17,… the formula of that arithmetic sequence is…
Solution :b = U2-U1
= 5-2= 3
Un= a+(n-1)b= 2 +(n-1)3= 2 + 3n-3= 3n+2-3= 3n-1
BARISAN BILANGAN GEOMETRI
Geometric Sequence Number
Geometric sequence
A geometric sequence is also called measurement sequence.
A geometric sequence is defined as a sequence which achieved by multiplying previous terms with constant number which is ≠ 0. this constant number is called as ratio, and its notation is r.
In geometric sequence of:U1 , U2 , U3 , . . . . . , Un-1 , Un it prevails r=
U2=U3=U4=…….= Un
U1 U2 U3 Un-1
for r is ratio and n is natural numbers..
Formula
Geometric sequence can be written as follow:a, ar, ar2, ar3,
U1 U2 U3 U4
The formula of geometric sequences for the n-term is given as follow:
Un = arn-1 Where:Un = the n-terms, for n is natural numbers
a = the 1st term (U1)
r = ratio
Geometric sequences can be determined by seeing the value of ratio (r).
If r > 1, it means that the geometric sequence is up.If 0 > r > 1, it means that the geometric sequence is down.
Example:1. Find the 6th term of the sequence 2, 6, 18 . . . .2. Find the ratio of geometric sequence if a = 27 and
U4 = 1
Solution :3. We know, a = 2 and U2 = 6r = Un = U2 = U2 = 6 = 3
Un-1 U2-1 U1 2
Thus,Un = arn-1
U6 = ar6-1 = ar5 = 2 . 35 = 2 . 243 = 486
Hence, the 6th term of the sequence 2, 6, 18 . . . Is 486
exAmpLe
Example :
2. We know, a = 27 and U4 = 1Un = arn-1 U4 = ar4-1 = ar3
1 = 27r3
r3 = 1 = 1 27 3
r = 1 3
Thus, the ratio value is 1 3 exAmpL
e
1. Find the ratio value and the 5th term of the following geometric sequences…
a. 2, 6, 18, 54b. 81, 27, 9, 3c. 72, -36, 18, -9d. 2, -4, 8, -16
Do this exercise NOW !!!
DERET ARITMATIKA
Arithmetic Series
Introduce
Jika pada barisan aritmatika, kita menggunakan koma untuk
membedakan satu suku dengan suku lainnya. Tapi pada deret aritmatika kita
mengggunakan tanda tambah.
Introduce
Dalam deret aritmatika kita akan menggunakan istilah-istilah : Suku 1 = asuku 2 = a + bsuku 3 = a + 2bDalam barisan aritmatika kita menggunakan rumus :a + (n-1 )b tapi dalam deret aritmatika rumus tersebut kita gunakan sebagai awal mula dari rumus suku ke-n dalam deret aritmatika.
Pay Attention to this formula
Deret pertamaSn = a + (a+b) + (a+2b) + (a+3b) + ……… + {a+(n-1)b}
Supaya kita menemukan rumus deret aritmatika, kita harus membalikkan deret di atas.
Deret keduaSn = {a+ (n-1)b} +………. + (a+3b) + (a+2b) + (a+b) + a
Tahap kemudian kita tambahkan deret pertama dengan deret kedua.
Sn = a + (a+b) + ……… + {a + (n-2) b } + {a + (n-1) b }Sn = {a+ (n-1)b} + {a+ (n-2) b} + ……... + (a+b) + a
2Sn = {2a + (n-1)b} + { 2a+ (n-1)b} + ……. + {2a+ (n-1)b} + {2a + (n-1)b}
2Sn = n { 2a + (n-1) b }
Sn = n { 2a + (n-1) b }
2 Sn = n { a + a + (n-1) b }
2 Sn = n (a+ Un )
2
Pay Attention to this formula
+
Pay Attention to this formula
yeah !!!!!!wufh…wufh….wuhf….
uh…….
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certificate for all of student
Type of pattern The formula
The n- term of odd numbers 2n-1
The sum of odd numbers in sequence
n²
The n-term of even numbers 2n
The sum of even number in sequence
n(n+1)
The n-term of triangle numbers n(n+1)
2
The n-term of square numbers n²
The n-term of rectangle numbers n(n+1)
The n-term of Pascal triangle 2n-1