Uploaded by Cecile Calunsod Tiburan

Report on Sequences

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Republic of the Philippines
State Universities and Colleges
GUIMARAS STATE COLLEGE
Buenavista, Guimaras
GRADUATE STUDIES
Master of Arts in Teaching Mathematics
SELECTED TOPICS IN MATHEMATICS
MATH 217
DR. ETHEL P. JUNCO
Professor
CECILIA T. BESARES
Reporter
Topic:
SEQUENCES, SERIES
AND
PROGRESSIONS
On this topic, you should be able to:
• recognise the difference between a sequence and a
series;
• recognise an arithmetic progression;
• find the n-th term of an arithmetic progression;
• find the sum of an arithmetic series;
• recognise a geometric progression;
• find the n-th term of a geometric progression;
• find the sum of a geometric series;
• recognize a harmonic series , harmonic mean.
• find the sum of the terms of harmonic series
• recognize a Fibonacci Sequence
3
What is a sequence?
It is a set of numbers which are
written in some particular order. We
sometimes write 𝑎1 for the first term, 𝑎2 for
the second term, and so on. We write 𝑛𝑡ℎ
term as 𝑎𝑛 .
A sequence is a function whose domain
is the finite set {1,2,3,…..,n} or the infinite
set {1,2,3,…}.
Example 1:
n
1
2n
2
2
4
3
6
4
8
5
10
The finite sequence has 5 terms.
We may use the notation
𝑎1 , 𝑎2 , 𝑎3 ,…., 𝑎𝑛
to denote a(1) , a(2) , a(3) , ….., a(n) , respectively.
Example 2:
n
1
𝒂𝒏
𝟒
2 3 4 …
𝟕 𝟏𝟎 𝟏𝟑 …
The above sequence is an infinite sequence
where 𝒂𝒏 = 3n + 1
Deriving the Rule
for Generating a Sequence
7
A.
4, 5, 6, 7, …
Solution:
a1 = 3 +1 = 4
a2 = 3 + 2= 5
Equation:
an = 3 + n or n + 3
a3 = 3 + 3 = 6
a4 = 3 + 4 = 7
B.
𝟏
𝟐
,
𝟒
𝟓
,
a1 =
1
1+1
=
1
2
a2 =
2
2+1
=
2
3
a3 =
3
3+1
=
3
4
,
𝟐
𝟑
,
𝟑
𝟒
𝟓
𝟔
,…
Solution:
Equation:
an =
𝒏
𝒏+𝟏
a4 =
4
4+1
a5 =
5
5+1
=
4
5
=
5
6
C.
1,3,5,7,…
Solution:
a1 = 2(1) -1= 1
a2 = 2(2) -1= 3
Equation:
an = 2n - 1
a3 = 2(3) -1 = 5
a4 = 2(4) -1 = 7
SEATWORK
Write an equation that describes the sequence.
1) 1 , 2 , 4 , 8 , …
2) -1 , 1 , 3 , 5 , …
3) x + 2 , 2x + 1, 3x , …
4)
1
2
, , ,
1
4
1
8
1
,…
16
5)
1
2
,1,
3
2
,2,…
12
Arithmetic Sequence
What is an Arithmetic Sequence?
It is a sequence in which each term
after the first is obtained by adding a
constant d to the preceding term.
The constant d is called the common
difference.
The nth term of an arithmetic sequence is defined
as
an =a1 + (n − 1)d
where a1 = the first term
an = the nth term
d = the common difference
Example A
Find the tenth term of the arithmetic sequence:
32, 47, 62, 77,...
d = 47 – 32 =15
a1 = 32
an =a1 + (n − 1)d
a10 = 32 + (10 − 1)15
a10 = 167
Example B
The fourth term of the arithmetic sequence is 18 and the
sixth term is 28. Give the first 3 terms:
a4 = 18 a6 = 28
Assumethata3 as a1
and a6 as a3.
an =a1 + (n − 1)d
a3 =a1 + (3 − 1)d
28 =18 + (3 − 1)d
28 = 18 + (2)d
2d = 10
d=5
Find:
a3 , a2 , a1
Using d = 5 and a4 = 18
a3 = 18 – 5 = 13
a2 = 13 – 5 = 8
a1 = 8 – 5 = 3
Example C
Find the number of terms in the arithmetic sequence:
50, 47, 44, …., -34
d = 47 – 50 = -3
an =a1 + (n − 1)d
−34 =50 + (n − 1)−3
−34 =50 −3n − 3
3n = 87
n = 29
There are 29 terms.
Arithmetic Series
A series is the indicated sum of the terms of a
sequence.
An algebraic expression of the form
a1 + (a1 + d) + (a1 + 𝟐d) + … + [a1 + (𝐧 − 𝟏)d]
is called an arithmetic series.
The formula for the sum of the first n terms in an
arithmetic sequence is
n
Sn= (a1+an)
2
or
n
Sn= [2a1+ (n − 1)d]
2
where Sn = sum
a1 = first term
an = the nth term
d = the common difference
Example A
Find the sum of the first ten terms of the arithmetic sequence
4, 10, 16, ...
Solutions: a1 = 4 ; n = 10 ; d = 6 ; S10 = ?
n
Sn= [2a1+ (n − 1)d]
2
10
[2(4)+ (10 − 1)6]
2
= 5[8 + 9(6)]
= 5(8+54)
= 5(62)
S10= 310
S10=
Example B
The sum of five consecutive numbers is 100. Find the first number
Solution: S5 = 100 ; d = 1; a1 = ?
n
Sn= [2a1+ (n − 1)d]
2
5
S5= [2(a1)+ (5 − 1)1]
2
5
100 = [2a1 + 4]
2
5
2
(100) = 2a1+4
40 = 2a1+4
2a1= 36
a1 = 18
The sequence is 18, 19, 20, 21, 22.
Example C
Let an be an arithmetic progression, for which a1 =15 and d = 3.
Find the sum of the first 10 elements.
Solution: a1 = 15
d=3
S10= ?
n
Sn= [2a1+ (n − 1)d]
2
10
S10=
[2(15)+ (10 − 1)3]
2
= 5[30 + (9)3]
= 5 [57]
= 285
S10 = 285
Arithmetic Mean
Is the average of a set of numerical values,
calculated by adding them together and dividing by
the number of terms in the set.
a+b
m=
2
TEST
Answer the following.
1.
2.
Find the 23rd term of an arithmetic sequence with first term 2
and common difference 7.
Find the sum of the first five terms of the arithmetic
sequence with first term 3 and common difference 5.
3-4. Write down the 10th and 19th terms of the arithmetic
sequence
a. 8, 11, 14, …
b. 8, 5, 2, …
2b b
, , 0,...
5-6. An arithmetic sequence is given by b,
3 3
a. State the sixth term
b. If the 20th term has a value of 15, find b.
7.
8.
After knee surgery, your trainer tells you to return to your jogging
program slowly.
He suggests jogging for 12 minutes each day for the first week.
Each week thereafter, he suggests that you increase that time
by 6 minutes per day.
How many weeks will it be before you are up to jogging 60
minutes per day?
You visit the Grand Canyon and drop a penny off the edge of
a cliff. The distance the penny will fall is 16 feet the first second,
48 feet the next second, 80 feet the third second, and so on in
an arithmetic sequence. What is the total distance the object
will fall in 6 seconds?
9.
A shop assistant is arranging a display of a
triangular array of tins so as to have one tin in the
top row, two in the second, three in the third and
so on. If there are 66 tins altogether, how many
rows can be completed arranged?
10. A restaurant has square tables which seat four people.
When two tables are placed together, six people can
be seated.
If 20 square tables are placed together to form one
long table, how many people can be seated?
If 1000 square tables are placed together to form one
very long table, how many people can be seated?
1.
Find the 23rd term of an arithmetic sequence with first term 2
and common difference 7.
(156)
2.
Find the sum of the first five terms of the arithmetic
sequence with first term 3 and common difference 5. (65)
3-4. Write down the 10th and 19th terms of the arithmetic
sequence
a. 8, 11, 14, … (35, 62)
(-19, -46)
b. 8, 5, 2, …
2b b
5-6. An arithmetic sequence is given by b,
, , 0,...
3 3
a. State the sixth term(-2b/3)
b. If the 20th term has a value of 15, find b.
(-45/16)
Geometric
Sequence
32
A geometric sequence is a sequence in
which a term is obtained by multiplying the
preceding term by a constant number, called
the common ratio, r.
The nth term of an geometric sequence is
an = a1r n−1
where a1 = the first term
an = the nth term
r = the common ratio
Example
Write down the 8th term in the Geometric Progression 1, 3, 9, ...
an = a1r n−1
Example
Find the number of terms in the geometric progression 6, 12, 24, ..., 1536
The formula for the sum of the first n terms in an
geometric sequence is
a1 − a1r n
Sn =
1 −r
or
a1 (1 − r n )
Sn =
1 −r
where Sn = sum
a1 = first term
r = the common ratio, r ≠ 1.
Example
Find the sum of each of the geometric series
Infinite geometric series is the indicated sum of
terms of an infinite geometric sequence.
For example: 1 + 2 + 3 + 4 + 5 + 6 + 7 + …
The sum of the terms in an infinite geometric
sequence is
a1
S=
1 −r
where a1 = first term ; r = the common ratio, |r| < 1.
Example
Find the sum of
S=
S=
a1
1 −r
3
10
a1 =
3
10
1
10
1−
=
3
100
+
+
3
10
3
10
*
3
1000
r=
10
9
=
+…
1
10
1
3
𝐒=
𝟏
𝟑
Example
Find the sum of 64 + 32 + 16 + …
a1
S=
1 −r
S=
64
1
1 −2
a1 = 64
=
64
1
2
= 128
r=
32
64
1
=2
𝐒 = 𝟏𝟐𝟖
Example
Express 0.212121… as a fraction in lowest terms.
0.212121… = 0.21 + 0.0021 + 0.000021 + …
S=
S=
a1
1 −r
0.21
1 −0.01
a1 = 0.21
=
0.21
0.99
=
r=
7
33
0.0021
0.21
= 0.01
𝐒=
𝟕
𝟑𝟑
TEST
a) Find a 20 given that a 3 = 1/2 and a 5 = 8
b) Find a 30 given that the first few terms
of a geometric sequence are given by
-2 , 1 , -1/2 , 1/4 ...
c) Find r given that a 1 = 10 and a 20 = 10 -18
d) write the rational number 0.9717171... as a
ratio of two positive integers.
ANSWERS
a) a 20 = 2 18
b) a 30 = 1 / 2 28
c) r = 0.1
d) 0.9717171... = 481/495
Geometric Mean
are the terms between any two nonsuccessive
terms in geometric sequence.
m= 𝑎𝑏
m= − 𝑎𝑏
if a and b are positive
if a and b are negative
Harmonic
Sequence
51
A sequence of quantities whose
reciprocals form an arithmetic
progression is said to be harmonic.
It may be represented as
1
a1
+
1
a1 + d
+
1
a1+ 2d
+…+
1
a1+ (n − 1)d
Find the next three terms in the following
harmonic progressions:
1
1
1
1
1) , , ,
,…
2 5 8 11
2) 19 , 17 , 15 , 13, …
3) 12 , 6 , 4 , 3 , …
Harmonic Series
1
1 1
1
1
S= 1 +
+ +
+
+ …+
+ …
2
3 4
5
𝑛
It has an infinite number of terms.
Example
Find the sum of the first 10 terms of the harmonic series.
1
1 1
1
S= 1 +
+ +
+
2
3 4
5
2520 1260 840
S=
+
+
2520 2520 2520
420
360
315
+
+
+
2520 2520 2520
1 1 1
1
1
+ + +
+ +
6 7 8
9 10
630
504
+
+
+
2520 2520
280
252
𝟕𝟑𝟖𝟏
+
=
2520 2520 2520
The sum of the first 10 terms of the harmonic series is
𝟕𝟑𝟖𝟏
2520
Find the sum of the first n terms of the
harmonic progressions when:
1) 𝑛 = 11
2) 𝑛 = 2
3) 𝑛 = 5
Harmonic Mean
A harmonic mean of two numbers is
the reciprocal of an arithmetic mean
of the two numbers.
2ab
H=
a+b
Fibonacci
Sequence
58
Fibonacci Sequence
Generalized sequence of first two
positive integers and the next
number is the sum of the previous
two, i.e. 1,1,2,3,5,8,13,21,…
The Rule is
xn = xn-1 + xn-2
where:
xn is term number "n"
xn-1 is the previous term (n-1)
xn-2 is the term before that (n-2)
Example: term 9 is calculated like this:
x9= x9-1 + x9-2
= x8 + x7
= 21 + 13
= 34
Thank You!
62
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