1. SEQUENCE FUNCTION
Sequence is a set of numbers written in a given order.
Function whose domain is the set of natural numbers and whose range is a
subset of the real numbers
Terms are the numbers or values in the sequence
Examples:
1. 1,2,3,4,5,6,...,n
2. 2,4,6,8,10,…,n
3. 5,10,15,20,…,n
A Sequence Function as a function whose domain is a subset of positive integers.
Example:
A ball is dropped from the height of 10 feet. Each time that it bounces, it reaches
a height that is half of its previous height. We can list the height to which the ball
bounces in order until it finally comes to rest.
After Bounce 2 4 6 5 8
Height (ft) 2 5 7 6 5
The numbers 2,4,6,8,10 form a sequence which is also an example of a sequence.
It can be in a form of ordered pairs and we let x as the after bounce and y as the height.
{(2,2),(4,5),(6,7),(5,6),(8,5)}
Finite Sequence is a function whose domain is the set of integers and has the last
term {1,2,3,…,n}. Infinite sequence is a list that continues without ends.
Example:
Identify whether the sequence is finite or infinite.
1. The first 5 natural numbers: 1, 2, 3 ,4, 5
2. The set of even whole numbers: 2, 4, 6, 8 , 10 ...
3. The multiples of 4 up to the number 40: 4, 8, 12, 16,…, 40.
4. The sequence of the multiples of 3: 3, 6, 9, 12...
Solution:
1 and 3 are examples of finite sequence while the examples in 2 and 4 are infinite
sequence. The 3 dots at the end of the infinite sequence indicates that the sequence goes
on without stopping.
FINDING THE TERM IN THE SEQUENCE
Sequence is usually denoted by their general or nth term.
For example,
푡푛 =
푛 + 2
푛
2. The equation provides how we will be able to compute for the nth term of the
sequence. To get the value of a specific term , replace n with the number of the term.
For example:
To get the first 3 terms of 푡푛 = 푛+2
푛
, replace n with 2,4 and 6.
Solution:
푡푛 =
푛 + 2
푛
=
2 + 2
2
= 2
푡푛 =
푛 + 2
푛
=
4 + 2
2
= 3
푡푛 =
푛 + 2
푛
=
6 + 2
2
= 4
The domain of the sequence is the set {2,4,6}.
The range is denoted as {2,3,4}.
Example:
1. Find the first 5 terms of the sequence:
푡푛 = 2푛 − 3
The domain is the set {1, 2, 3, 4, 5}
Substituting this into 푡푛 = 2푛 − 3, we have:
푡푛 = 2(1) − 3 = -1
푡푛 = 2(2) − 3 = 1
푡푛 = 2(3) − 3 = 3
푡푛 = 2(4) − 3 = 5
푡푛 = 2(5) − 3 = 7
The first 5 terms of the sequence 푡푛 = 2푛 − 3 is {-1,1,3,5,7}
2. Find the first 3 terms of the sequence:
푡푛 = 푛2
The domain is {1,2,3}
Substituting this into 푡푛 = 푛2, we have:
푡푛 = 12 = 1
푡푛 = 22 = 4
푡푛 = 32 = 9
3. The first 3 terms of the sequence {1,4,9}
FINDING THE GENERAL TERM
When given the first few terms of the sequence, we can look at the pattern in order to
obtain the next terms or a general statement regarding the nth term.
Example:
1. Find the next term and describe the pattern.
1,4,9,16,25…
The next term is 36. The terms are all squares.
1st Term 12= 1
2nd Term 22= 4
3rd Term 32= 9
4th Term 42= 16
5th Term 52= 25
6th Term 62= 36
SERIES
Series is the sum of the terms
t1 + t2 + t3 + t4 +,…, + tn
Note: The sum of the infinite sequence is an infinite series. A partial sum is the sum of
the first n terms. A partial sum is also called the sum of finite series, and is denoted as sn ,
where n denotes the number of terms in the sum.
For example:
If 4, 7, 10, 13, 16, 19…. Is a sequence, then 4 + 7 + 10 + 13 + 16 +19 is the
corresponding sum or series
If only we want to get the sum of only 4 terms in the sequence, it is a partial sum
written as S4 , which is equal to 4 + 7 + 10 + 13.
Example:
For the sequence 9, 13, 17, 21, 25, 29, 33, 37, find
1. S4
2. S1
Solution:
1. S4= 9+13+17+21 = 60
2. S1=9+13+17+21+25+29+33 = 147
4. SIGMA NOTATION
Use the symbol Σ, a capital sigma of the Greek alphabet which corresponds to
the letter S.
The symbol i ,is called the index of summation.
Used as summation or the total sum of the sequence.
Example:
1
5
Σ 푖 2
푖=1
= 12 + 22 + 32 + 42 + 52
What we have observed here is that, the i =1 is located under the sigma symbol.
So, it indicates that the first term on the right-hand side is the value of i2 when i is equal
to 1. The next terms are the values of i2 when i is 2,3,4 and 5. We stop here because in the
upper sigma notation there is number 5.
The sigma notation can be defined by the equation:
푛
Σ 퐹(푖) = 퐹(푚) + 퐹(푚 + 1) + 퐹(푚 + 2) + ⋯ + 퐹(푛)
푖=푚
Where m and n are integers and m ≤ n. The right hand sign of the equation consist of the
sum of n – m + 1terms, the first of which is obtained by replacing i by m + 1 and so on
until the last term is obtained.
The upper limit will be m
The lower limit will be n
The symbol i is the “dummy” symbol because any other symbol could be used
without changing the right side.
Example:
7
Σ 푖3
푖 =2
= 23 + 32 + 43 + 53 + 63 + 73
Sometimes the term of a sum involve subscripts . For instance the sum
푛1 + 푛2 + 푛3 + 푛4 + ⋯ + 푛푛
Can also be written in sigma notation
푛
Σ 푛푖
푖 =1
5. A. For each sequence, find the indicated partial sum.
1. 3,5,7,9,11…S5
2. 2,4,8,16,32…S7
3. 1,2,3,4,5,6,7,8,9,10,11,12,13,14,…,S6
4. 5,10,15,20,25,30,35,40,…,S7
B. Find the indicated term of the sequence.
1.푡푛 = 4푛 − 7: 푡5
2. 푡푛 = 4푛−2
3
: 푡6
3. 푡푛 = 5 + 2푛: 푡10
4. 푡푛 = 4푛
2
: 푡4
5. 푡푛 = 4(푛 − 7)2: 푡5
c. write the following in sigma notation
1. 2+4+6+8+10
2. 1+3+5+7+9+11+13+15+17
3. 52+62+72
4. 1+
1
2
+ 1
3
+ 1
4
+ 1
5
5. (n+1) + (n+2) +n+3)… (n+n)
PRINCIPLE MATHEMATICAL INDUCTION
Definition Let P(n) be a proposition on an integer variable n. Then P(n) is true for
all integers n s if and only if the following two conditions are both satisfied :
(i) P(s) is true ,
ii) If P(k) , where k s , is assumed to be true, then P(k 1) is true .
6. Example:
We use the mathematical induction to prove that
1+3+5+7+….+(2n – 1) =n2
STEP 1. We apply that the formula is true for n = 1. If n = 1 then the formula becomes
1=12
1=1
which is true.
STEP 2. Show that Pk is true.
1 + 3 + 5 + 7 + ⋯ + (2푘 − 1) = 푘2
If the equation 1 is true, then
1 + 3 + 5 + ⋯ + (2푘 − 1) + [2(푘 + 1) − 1] = (푘 + 1)2
This is also true.
Then we add 2k +1 – 1 to the left side of the Equation 1 and its equivalent 2k+1 to the
right side and we have:
1 + 3 + 5 + ⋯ + (2푘 − 1) + [2(푘 + 1) − 1] = 푘2 + (2푘 + 1)
1 + 3 + 5 + ⋯ + (2푘 − 1) + [2(푘 + 1) − 1] = (푘 + 1)2
Which is Equation 2.
Example:
Use mathematical induction to prove that n3 + 3n2 + 2n is divisible by 3 for n 1.
Basis case: n = 1 n3 + 3n2 + 2n = 13 + 3 ·12 + 2·1 = 1 + 3 + 2 = 6 and 3 | 6.
Hypothesis: Assume k3 + 3k2
(i.e. k3 + 3k2 + 2k = 3s for some integer s.)
Induction step:
(k + 1) 3 + 3(k + 1)2 + 2(k + 1) = (k3 + 3k2 + 3k + 1) + 3(k2 + 2k + 1) + 2(k + 1)
= (k3 + 3k2 + 2k) + k + 1 + 3k2 + 6k + 3 + 2k + 2
= 3s + 3k2 + 9k + 6 hypothesis and algebra
= 3s + 3(k2 + 3k + 2) factoring
= 3[s + k2 + 3k + 2] factoring
Therefore 3 | [(k + 1)3 + 3(k + 1)2 + 2(k + 1)].
Therefore, by the principle of mathematical induction, the statement is true for all n.
Example:
Use mathematical induction to prove that 2n < n! for n ≥ 4
Basis: n = 4 lhs 24 = 16 rhs 4! = 24 Since 16 < 24 the basis holds.
7. Hypothesis: Assume 2k < k! for k ≥ 4
Induction step:
Proof: 2k+1 = 2 • 2k
< 2 • k!
< (k + 1) • k! since k >=4
= (k + 1)!
Therefore, by the principle of mathematical induction, the statement is true for all n ≥ 4.