# Sequences and Series - PowerPoint by zhangyun

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```									  Arithmetic and
Geometric Sequences
and their Summation
14.1 Sequences
Find the next two terms of the following sequences :
(1) 2, 5, 8, 11,…… arithmetic sequence
(2) 2, 6, 18, 54, …. geometric sequence
(3) 2, 4, 8, 16,……. geometric sequence
(4) 5, -25, 125, -625, …. geometric sequence
(5) 3, 4, 6, 9, 13, …….
(6) 5, 2, -1, -4, …..   arithmetic sequence
(7) 0, sin20o, 2sin30o, 3sin40o
14.1 Sequences
Consider the following sequence:
1, 3, 5, 7, 9, ….., 111
1 is the first term of the sequence,mathematically,
T(1) = 1 or T1 = 1
3 is the second term of the sequence, mathematically,
T(2) = 3 or T2 = 3
5 is the third term of the sequence, mathematically,
T(3) = 5 or T3 = 5
111 is the nth term of the sequence, mathematically,
T(n) = 111 or Tn = 111
14.1 Sequences
Consider the sequence
2, 4, 8, 16, ….
The sequence is formed from
timing 2 to the previous term.
So, the sequence can be represented by
the general term
T(n) = 2n or Tn = 2n
P.159
Ex. 14A
14.2 Arithmetic Sequence

An arithmetic sequence
(A.S. /A.P.) is a sequence
having a common difference.
14.2 Arithmetic Sequence
Illustrative Examples
14.2 Arithmetic Sequence
14.2 Arithmetic Sequence
14.2 Arithmetic Sequence
P.166
Ex. 14B
14.2 Arithmetic Sequence
Arithmetic Means
When a, b and c are three consecutive
terms of arithmetic sequence, the
middle term b is called the arithmetic
mean of a and c.
ac
b
2
14.2 Arithmetic Sequence
Arithmetic Means
Insert two arithmetic means between 11 and 35.
T (1)  a  11.......... 1)
....(
T (2)  a  d .......... 2)
.....(
T (3)  a  2d .......... 3)
...(
T (4)  a  3d  35.....(4)
14.2 Arithmetic Sequence
Insert two arithmetic means between 11 and 35.

(4)  (1)       3d  24
d 8
1st arithmetic mean  a  d  19
2nd arithmetic mean  a  2d  27
P.170
Ex. 14C
14.3 Geometric Sequence

A geometric sequence
(G.S. / G.P.) is a sequence
having a common ratio.
14.3 Geometric Sequence
Illustrative Examples
14.3 Geometric Sequence
14.3 Geometric Sequence
14.3 Geometric Sequence
P.176
Ex. 14D
14.3 Geometric Sequence
Geometric Means
When x, y and z are three
consecutive terms of geometric
sequence, the middle term y is called
the geometric mean of x and z.

y   xz
14.3 Geometric Sequence
Geometric Means
Insert two geometric means between 16 and -54.

T (1)  a  16.......... 1)
....(
T (2)  aR..........        2
.........( )
T (3)  aR ..........
2
3
.......( )
T (4)  aR  54......( )
3
4
14.3 Geometric Sequence
Insert two geometric means between 16 and -54.
( 4)          54  27
R  3

(1)           16     8
3
R
2
1st goemetric mean  aR   24
2nd goemetric mean  aR  36 2
P.181
Ex. 14E
14.4 Series
Let’s consider a sequence :
T(1), T(2), T(3), T(4), …., T(n)
The expression T(1) + T(2) + T(3) +….+ T(n)
is called a series. We usually denote the sum
of the first n term of a series by the notation
S(n).
 S (n)  T (1)  T (2)  T (3)  .... T (n)
14.5 Arithmetic Series

Arithmetic Sequence : 2, 5, 8, 11, …

Arithmetic Series : 2 + 5 + 8 + 11 + ….
14.5 Arithmetic Series
Formula of Arithmetic Series

S(n) = a + a + d + a + 2d +
a + 3d + …. + a + (n - 1)d
l
14.5 Arithmetic Series
Formula of Arithmetic Series

S(n) = l + l - d + l - 2d +
l - 3d + …. + a + d + a
14.5 Arithmetic Series
S(n) = a + a + d + a + 2d + a + 3d + ………... + a + (n - 1)d

S(n) = l + l – d + l - 2d + l - 3d + ….+ a + d + a

2S(n) =(a + l)+(a + l)+(a + l)+(a + l)+….. +(a + l)
2S(n) = n(a + l)
n( a  l )
S ( n) 
2
14.5 Arithmetic Series
n( a  l )
S ( n) 
2
n
S (n)  [a  a  (n  1)d ]
2
n
S (n)  [2a  (n  1)d ]
2
P.189
Ex. 14F
14.6 Geometric Series

Geometric Sequence : 3, 9, 27, 81, …

Geometric Series : 3 + 9 + 27 + 81
14.6 Geometric Series

Formula of Geometric Series

S(n) = a + aR +     +  aR2   aR3

+ …. + aRn-1
14.6 Geometric Series
Formula of Geometric Series

R.S(n) = aR +     aR
+  2   aR 3   +
aR4 + …. + aRn
Subtracting two series
S(n) = a + aR + aR2 + aR3 + …. + aRn-1
R.S(n) = aR + aR2 + aR3 + aR4 + …. + aRn

S(n) –R.S(n) = a - aRn
(1 – R) S(n) = a (1 – Rn)

a(1  R )
n
S ( n)            , where R  1
1 R
14.6 Geometric Series
a(1  R )n
S ( n)            , where R  1
1 R
Timing –1 on both numerator and denominator

a( R  1)
n
S ( n)            , where R  1
R 1
P.196
Ex. 14G
14.6 Geometric Series

Sum to Infinity of a Geometric Series
14.6 Geometric Series
Sum to Infinity of a Geometric Series
Consider such a Geometric Series
1 1 1 1
      .....
2 4 8 16
What is the value of common ratio R ?
1
R
2
14.6 Geometric Series
Sum to Infinity of a Geometric Series
1
R
2
Consider Rn where n tends to the infinity
1 2          1 3            1 4
( )  0.25   ( )  0.125    ( )  0.0625
2            2              2

1 5           1 6            1 7
( )  0.03125 ( )  0.015625 ( )  0.0078125
2             2              2
1 
What will occur for   ( )    if n tends
2
to infinity ?
1 
( ) 0
2

R       0
where –1< R <1
Summation of a geometric Series to infinity

a(1  R )
S ( )            , where  1  R  1
1 R

a
S ( )       , where  1  R  1
1 R
P.203
Ex. 14H
(extension module)

Summation Notation


4

Consider the symbol         T (r )
r 1

where T( r ) = 3r + 5

4

 T (r )
r 1
= 3(1) + 5 + 3(2) + 5 +3(3)
+ 5 + 3(4) +5
= 50

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