Understanding Sequences and Series

 
SEQUENCES AND
SERIES
 
Introduction - Sequence
 
A sequence or progression is an ordered set of
numbers which can be generated from a rule.
General sequence terms as denoted as follows
a
1 
– first term
 
, a
2
 – second term, …, a
n
 – n
th
 term etc
The rule may give n
th
 term, a
n
, as a function of n
 
Introduction – Sequence (cont)
 
Example: Given a infinite sequence
 
 
The rule or general term
 
Example: Given a infinite sequence
 
 the general term is
 
 
Exercises:
 
i.
Write down the first four term of the sequence
with general term
 
ii.
What is the twenty-first term of the sequence?
iii.
What is the 100
th
 term of the sequence of
 
Arithmetic Sequence (Progression)
-AP
 
An arithmetic sequence or arithmetic progression
is a sequence of numbers such that the difference
of any two successive members of the sequence
is a common constant.
In general, an AP is written in the form of
      {a, a + d, a + 2d, a + 3d,…a + (n – 1)d,…}
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c
e
 
Arithmetic Sequence (Progression) –
AP (cont)
 
If the initial term of an AP is a
1
 and the common
difference of successive numbers is d, then the n
th
term of the sequence is given by:
 
Example 1
: Write down the n
th
 term of the
arithmetic sequences
a)
 -10, -5, 0, 5,…
b)
 
 
 
Arithmetic Sequence (Progression) –
AP (cont)
 
Example 2
:
   The 4
th
 term of an arithmetic sequence is 12 and
the tenth term is 42.
a)
Given that the first term is u
1
 and the common
difference is d, write down two equations in u
1
and d that satisfy this information.
b)
Solve the equations to find the values of u
1
 and
d.
 
Geometric Sequences (Progression) –
GP
 
A geometric progression (GP) is 
a sequence of
numbers in which each number is multiplied by the
same factor to obtain the next number in the
sequence.
For example, the sequence 2, 6, 18, 54,… is a GP
with common ratio 3.
Thus, the general form of a GP is
 
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Geometric Sequences (Progression) –
GP (cont)
 
The nth term of a GP with initial value a and
common ration r is given by
 
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o
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e
:
 
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       If r is 0, then we have : a, 0, 0, 0, …   (not GP)
       If r is 1, then we have : a, a, a, a, …   (not GP)
Example 3
: Write down the first five terms of the
GP which has first term 1 and common ration ½ .
Find the 10
th
 and 20
th
 term of the sequence.
 
Geometric Sequences (Progression) –
GP (cont)
 
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/
8
.
 
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1
.
 
Limit of infinite sequence
 
The limit of a sequence is the value to which its
term approach indefinitely as 
n becomes large
.
If the limit of a sequence a
n
 is L, we can write as
 
If a sequence has a (
finite
) limit, then it is said to
be 
convergent
.
If a sequence becomes 
arbitrarily large
(approaches 
), then it is said to be 
divergent
 
Limit of infinite sequence (cont)
 
Example 5
:
a)
b)
 
c)
 
d)
 
Series and Partial Sums
 
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.
 
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s
 
.
 
 
Arithmetic Series
 
Given an arithmetic sequence
 
Then, the arithmetic series can be written as
 
The sum of the first n terms of an arithmetic
sequence is
 
 
where a
1
 is the first term and 
l 
is a
n
 , the last term
of a finite sequence.
Or
 
Arithmetic Series (cont)
 
Example 6
:
a)
Find the sum of the first 50 terms of the
sequence {1, 3, 5, 7, 9, …}
b)
Find the sum of the series 1+ 3.5 + 6 + 8.5 + …+
101
 
Geometric Series
 
Given a geometric sequence
 
Then, the geometric series can be written as
 
The sum of the first 
n 
terms of an geometric
sequence is
 
 
provided that r 
 1.
 
Geometric Series (cont)
 
Example 7
:
a)
Given the first two terms of a geometric
progression as 2 and 4, what is the sum of the
first 10 terms?
b)
Find the sum of the first 20 terms of the
geometric series  2 + 6 + 18 + 54 + …
 
Convergence /divergence series
 
The n
th
 partial sum of an arithmetic series is
 
What will happen when n becomes very large
(approaches 
)
?
  That is
 
 The series approaches infinity.
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d
.
 
Convergence /divergence series
(cont)
 
The n
th
 partial sum of an geometric series is
 
What will happen when n becomes very large
(approaches 
)
 ?
  That is
 
If|r|<1, then r
n 
0 as n 

, thus the series is
converged i.e.
 
If|r| > 1, then r
n

 as n 

, thus the series is
diverged
 
Convergence /divergence series
(cont)
 
Example 8
:
a)
For the geometric progression whose first two
terms are 5 and ½ , find S
.
b)
Consider a geometric progression whose first
three terms are 12, -6, 3. Find both S
8
 and S
.
c)
For the geometric progression whose first two
terms are 2 and 4, find S
7
, S
20
 and S
.
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Exploring the concepts of sequences and series in mathematics, including definitions, examples, and exercises on arithmetic sequences, geometric progressions, and general terms. Learn about generating sequences, finding nth terms, common differences, and common ratios in different types of sequences. Practice solving problems and understanding the relationships between terms in a sequence or progression.


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  1. SEQUENCES AND SERIES

  2. Introduction - Sequence A sequence or progression is an ordered set of numbers which can be generated from a rule. General sequence terms as denoted as follows a1 first term, a2 second term, , an nthterm etc The rule may give nthterm, an, as a function of n

  3. Introduction Sequence (cont) Example: Given a infinite sequence 1 2, 2 3 ,3 4 ,4 5, The rule or general term ? ??= ? + 1 Example: Given a infinite sequence 1 2, 1 4 ,1 8 , the general term is 1 2? ??=

  4. Exercises: i. Write down the first four term of the sequence with general term ??= 2? 1 1 ii. iii. What is the 100thterm of the sequence of What is the twenty-first term of the sequence? ??= 1?? + 1 ?

  5. Arithmetic Sequence (Progression) -AP An arithmetic sequence or arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a common constant. In general, an AP is written in the form of {a, a + d, a + 2d, a + 3d, a + (n 1)d, } where a is the first term and d is the common difference

  6. Arithmetic Sequence (Progression) AP (cont) If the initial term of an AP is a1and the common difference of successive numbers is d, then the nth term of the sequence is given by: Example 1: Write down the nthterm of the arithmetic sequences a) -10, -5, 0, 5, b) ?, ? 2, 2,0,?

  7. Arithmetic Sequence (Progression) AP (cont) Example 2: The 4thterm of an arithmetic sequence is 12 and the tenth term is 42. a) Given that the first term is u1and the common difference is d, write down two equations in u1 and d that satisfy this information. b) Solve the equations to find the values of u1and d.

  8. Geometric Sequences (Progression) GP A geometric progression (GP) is a sequence of numbers in which each number is multiplied by the same factor to obtain the next number in the sequence. For example, the sequence 2, 6, 18, 54, is a GP with common ratio 3. Thus, the general form of a GP is ?,??,??2,??3, where a is the first term and r is the common ratio

  9. Geometric Sequences (Progression) GP (cont) The nth term of a GP with initial value a and common ration r is given by = n a 1 n ar Note: the common ratio r should not be 0 or 1 If r is 0, then we have : a, 0, 0, 0, (not GP) If r is 1, then we have : a, a, a, a, (not GP) Example 3: Write down the first five terms of the GP which has first term 1 and common ration . Find the 10thand 20thterm of the sequence.

  10. Geometric Sequences (Progression) GP (cont) Example 4: The 3rdterm of a GP is 3 and the 6th term is 3/8. Find the common ratio r and the first term a1.

  11. Limit of infinite sequence The limit of a sequence is the value to which its term approach indefinitely as n becomes large. If the limit of a sequence anis L, we can write as lim ? ??= ? If a sequence has a (finite) limit, then it is said to be convergent. If a sequence becomes arbitrarily large (approaches ), then it is said to be divergent lim ? ??=

  12. Limit of infinite sequence (cont) Example 5: a) b) lim ? 1 ? ? + 1 ? lim ? c) 1 lim ? ?2+ 1 d) ? ?2 lim

  13. Series and Partial Sums When we sum up just n terms of a sequence, it is called the partial sums Sn. ??= ?1+ ?2+ ?3+ + ?? When we sum up an infinite sequence, it is called a series . ??= ?1+ ?2+ ?3+ + ??+

  14. Arithmetic Series Given an arithmetic sequence ?,? + ?,? + 2?, ,? + ? 1 ?, Then, the arithmetic series can be written as ? = ? + ? + ? + ? + 2? + + ? + ? 1 ? + The sum of the first n terms of an arithmetic sequence is n Sn = + ( ) a l 2 where a1is the first term and l is an, the last term of a finite sequence. Or n a Sn ( 2 2 n = + ) 1 d

  15. Arithmetic Series (cont) Example 6: a) Find the sum of the first 50 terms of the sequence {1, 3, 5, 7, 9, } b) Find the sum of the series 1+ 3.5 + 6 + 8.5 + + 101

  16. Geometric Series Given a geometric sequence ?,??,??2, ,??? 1, Then, the geometric series can be written as ? = ? + ?? + ??2+ ??3+ + ??? 1+ The sum of the first n terms of an geometric sequence is 1 ( ) 1 n n ) ( a r a r = / S n 1 1 r r provided that r 1.

  17. Geometric Series (cont) Example 7: a) Given the first two terms of a geometric progression as 2 and 4, what is the sum of the first 10 terms? b) Find the sum of the first 20 terms of the geometric series 2 + 6 + 18 + 54 +

  18. Convergence /divergence series The nthpartial sum of an arithmetic series is ??=? 22? + ? 1 ? What will happen when n becomes very large (approaches )? That is ? 22? + ? 1 ? lim ? ??= lim ? The series approaches infinity. Thus, an arithmetic series is a divergence series for any a and d.

  19. Convergence /divergence series (cont) The nthpartial sum of an geometric series is ??=?(1 ??) 1 ? What will happen when n becomes very large (approaches ) ? That is ?(1 ??) 1 ? lim ? ??= lim ? If|r|<1, then rn 0 as n , thus the series is converged i.e. Sn =1 a r If|r| > 1, then rn as n , thus the series is diverged

  20. Convergence /divergence series (cont) Example 8: a) For the geometric progression whose first two terms are 5 and , find S . b) Consider a geometric progression whose first three terms are 12, -6, 3. Find both S8and S . c) For the geometric progression whose first two terms are 2 and 4, find S7, S20and S .

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