04. Sequence and Series-1

8/10/2019 04. Sequence and Series-1

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Sequences and Series

Session MPTCP04


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1. Finite and infinite sequences2. Arithmetic Progression (A.P.)

definition! nthterm". Sum of n terms of an A.P.4. Arithmetic Mean (A.M.) and

insertion of n A.M.s #et$een t$ogi%en num#ers.&. 'eometric Progression ('.P.)

definition! nthterm. Sum of n terms of a '.P.

Session #*ecti%es


3/63

Sequence + a ,efinition

A sequence is a function $hose

domain is the set - of naturanum#ers. _I001

a1! a2! a"! . . .! an! . . .


4/63

Finite and /nfinite Sequences

Finite sequence a a1! a2! a"! . . .! an

/nfinite sequence

a a1! a2! a"! . . .! an! . . .

_I001


5/63

Series + a ,efinition

/f

a1! a2! a"! . . .! an! . . .

is a sequence!

_I001

the eression

a13a23a"3 . . . 3an3 . . .

is caed a series.


6/63

Arithmetic Progression

A sequence is caed an arithmeticrogression (A.P.) if the difference#et$een an term and the re%iousterm is constant.

The constant difference! genera denoted # dis caed thecommon difference.

_I002

a15 a

a25 a3d

a"5 a3d3d 5 a32d

a45 a3d3d3d 5 a3"d

First term'enera Term

an5 a3d3d3d3... 5 a3(n1)d


7/63

/s a 'i%en Sequence an A.P.6

Agorithm to determine $hether a

gi%en sequence is an A.P.

Ste / #tain genera term an

Ste // ,etermine an31

#

reacing n # n31 in the genera term

Ste /// Find an31an. /f this is indeendent of n! the

gi%en sequence is an A.P.

_I002


8/63

Pro#em So%ing Ti

Choose Well!!!!

7 Terms Common diff.

" ad! a! a3d d

4 a"d! ad! a3d! a3"d 2d

& a2d! ad! a! a3d! a32d d

a&d! a"d! ad! a3d! a3"d! a3&d 2d

_I002


9/63

/ustrati%e ro#em

8. /f sum of three num#ers in A.P. is4&! and the second num#er isthrice the first num#er! find thethree num#ers.

_I002

A. 9et the num#ers #e ad! a! a3d'i%en that (ad)3a3(a3d) 5 4&

"a 5 4& a 5 1&

Aso! a 5 "(ad)

"d 5 "0

d 5 10

the three num#ers are &! 1&! 2&


10/63

/mortant Proerties of A.P.s

/f A a! a3d! . . .! a3(n1)d

adding constant : to each term!

$e get!

A; a3:! a3d3:! . . .! a3(n1)d3:

A; is aso an A.P. $ith the same common difference.

_I002


11/63


/f A a! a3d! . . .! a3(n1)d

mutiing each term # non


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a:

3an(:1)

5 a1

3an

: 5 2! "! 4! . . . (n1)

=ame

Consider A 2! 4! ! >! 10! 12! 14! 1! 1>! 20

_I002


13/63


=ame

Consider A 2! 4! 6! >! 10! 12! 14! 16! 1>! 20

a"3a>5 22

_I002a

:

3an(:1)

5 a1

3an

: 5 2! "! 4! . . . (n1)


14/63

/mortant Proerties of A.P.S

=ame

Consider A 2! 4! ! >! 10! 12! 14! 1! 1>! 20

a"3a>5 22 5 a&3a5 22

_I002a

:

3an(:1)

5 a1

3an

: 5 2! "! 4! . . . (n1)


15/63


=ame

Consider A 2! 4! ! >! 10! 12! 14! 1! 1>! 20

a"3a>5 22 5 a&3a5 22 5 a1+a105 22

_I002a

:

3an(:1)

5 a1

3an

: 5 2! "! 4! . . . (n1)


16/63


a! #! c are in A.P. 2# 5 a3c

_I002


17/63


A sequence is an A.P.

an5 An3?! A! ? are constants.

A is the common difference.

Proof

an5 a3(n1)d

or! an5 dn3(ad)

or! an5 An3?! $here A is the common difference

_I002


18/63


/f A a! a3d! . . .! a3(n1)d

ta:e e%er third term!

A; a! a3"d! a3d! . . . . . . . . . .

A; is aso an A.P.

_I002


19/63

Sum of n Terms of an A.P.

Sn5 a1 3(a13d)3 . . .3@a13(n2)d3@a13(n1)d

Aso!

Sn5 @a13(n1)d3@a13(n2)d3@a13d3. . .3a1

Adding!

2Sn5 n@2a13(n1)d

( ){ }n 1n

S 2a n 1 d2

= +

_I003


20/63

Sum of n Terms of an A.P.

( ){ }n 1n

S 2a n 1 d2

= +

This can aso #e $ritten as

( )

{ }= + +

n 1 1

nS a a n 1 d

2

{ }n 1 nn

S a a2

= +

{ }nn

S First Term 9ast Term

2

= +

_I003


21/63

Proert of Sum of n Terms of anA.P.

A sequence is an A.P.

Sn5 An23?n!

$here A! ? are constants.

2A is the common difference.

Be :no$ that! ( ){ }= + nn

S 2a n 1 d2

= +

2n

d dS n a n

2 2

earranging!

r! Sn5 An23?n.

_I003


22/63

Arithmetic Mean

A is the A.M. of t$o num#ers a and#

a! A! and # are in A.P.

Aa 5 #A

2A 5 a3#

_I004

a #A

2

+ =


23/63

Arithmetic Mean + a ,efinition

then A1! A2! A"! . . . ! Anare caed arithmetic means(A.M.s) of a and #.

4 2 2 4A1 A2 A" A4 A&

0a

#

/f n terms A1! A2! A"! . . . Anareinserted #et$een t$o num#ers aand # such that a! A1! A2! A"! . . . !

An! # form an A.P.!

_I004


24/63

Arithmetic Mean + Common,ifference

9et n A.M.s #e inserted #et$een t$onum#ers a and #

9et the common difference #e d

-o$ # 5 a3(n321)d 5 a3(n31)d

4 2 2 4

A1 A2 A" A4 A&

0

a

#

The A.P. thus formed $i ha%e (n32)terms.

_I004

m

# a # ad D A a m

n 1 n 1

= = +

+ +


25/63

Proert of A.M.s

9et n A.M.s A1! A2! A"! . . .! An#e

inserted #et$een a and #.

Then!

1 2 " n

a #A A A ... A nA n

2

++ + + + = =

_I004


26/63


27/63

'eometric Progression

Consider a fami $here e%er

femae of each generation has

eact 2 daughters.

/t is then ossi#e to determine the

num#er of femaes in each generation

if the generation num#er is :no$n.

1st'eneration 1 femae

2nd

'eneration 2 femaes"rd'eneration 4 femaes

Such a rogression is a 'eometric Progression ('.P.)

_I005


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Pro#em So%ing Ti

Choose Well!!!!

7 Terms Common ratio

" aGr! a! ar r

4 aGr"! aGr! ar! ar" r2

& aGr2! aGr! a! ar! ar2 r

aGr&! aGr"! aGr! ar! ar"! ar& r2

_I005


30/63


31/63

/mortant Proerties of '.P.s

/f ' a! ar! ar2! . . .! arn1

ta:ing reciroca of each term!

'; is aso a '.P. $ith a reciroca

common ratio.

2 n 1

1 1 1 1'H ! ! ! . . .!

a ar ar ar

_I005


32/63


/f ' a! ar! ar2! . . .! arn1

raising each term to o$er :!

'; a:! a:r:! a:r2:! . . .! a:r(n1):

'; is aso a '.P. $ith commonratio r:.

_I005


33/63


a:an(:1) 5 a1an

: 5 2! "! 4! . . . (n1)

=ame

Consider ' 1! 2! 4! >! 1! "2! 4! 12>! 2&! &12

_I005


34/63


=ame

Consider ' 1! 2! 4! >! 1! "2! 4! 128! 2&! &12

a"a>5 &12

_I005a:an(:1) 5 a1an

: 5 2! "! 4! . . . (n1)


35/63


=ame

Consider ' 1! 2! 4! >! 16! 32! 4! 12>! 2&! &12

a"a>5 &12 5 a&a5 &12

_I005a:an(:1) 5 a1an

: 5 2! "! 4! . . . (n1)


36/63


=ame

Consider ' 1! 2! 4! >! 1! "2! 4! 12>! 2&! 512

a"a>5 &12 5 a&a5 &12 5 a1a105 &12

_I005a:an(:1) 5 a1an

: 5 2! "! 4! . . . (n1)


37/63


a! #! c are in '.P. #25 ac

_I005


38/63


/f ' a! ar! ar2! . . .! arn1

ta:e e%er third term!

'; a! ar"! ar! . . .

'; is aso a '.P.

_I005


39/63


a1! a2! a"! . . . ! anis a '.P. of

ositi%e terms

oga1! oga2! oga"! . . . oganis anA.P.

_I005


40/63

Sum of n Terms of a '.P.

Sn5 a3ar3ar23ar"3 . . .3ar(n1) III(i)

Mutiing # r! $e get!

rSn5 ar3ar23ar"3 . . .3ar(n1)3arnII...(ii)

Su#tracting (i) from (ii)! (r1)Sn5 a(rn1)

( )( )

=

n

n

r 1S a

r 1

_I006


41/63

Cass =ercise 81.

8. /f og 2! og (2

1) and og (2

3")are in A.P.! find . _I002


42/63

Cass =ercise 81.8. /f og 2! og (21) and og (23") are in

A.P.! find .

_I002A. 'i%en thatog(21)og2 5 og(23")og(21)

x x

x

2 1 2 3log log

2 2 1

+ =

2x x 1 x 12 2 1 2 6+ + + = +2x x2 4.2 5 0 =

( ) ( )x x2 5 2 1 0 + =

( )x x2 5 2 cannot be negative = Q

= = 2log5

x log 5log2


43/63

Cass =ercise 82.

8. Sho$ that there is no infinite A.P.$hich consists on of distinctrimes. _I002


44/63

Cass =ercise 82.

_I002A. 9et! if ossi#e! there #e an A.P.consisting on of distinct rimes a1! a2! a"! . . .! an! . . .

an5 a13(n1)d

8. Sho$ that there is no infinite A.P. $hichconsists on of distinct rimes.

( )1a 1 1 1a a a 1 1 d+ = + +

1a 1 1a a (1 d)+ = +

Thus! (a131)thterm is a mutie of a1.

Thus! no such A.P. is ossi#e.

8.=.,.


45/63

Cass =ercise 8".

_I0038. ! $here Sndenotes

the sum of the first n terms of anA.P.! then common difference is

(a) P38 (#) 2P3"8

(c) 28 (d) 8

(J.=.=. Best ?enga 14)

( )nn

S nP n 1 Q2= +


46/63

Cass =ercise 8".

_I003

8. ! $here Sndenotes

the sum of the first n terms of an A.P.!then common difference is

(a) P38 (#) 2P3"8

(c) 28 (d) 8

(J.=.=. Best ?enga 14)

( )nn

S nP n 1 Q2

= +

A. an5 Sn Sn1

( ) ( ) ( )

( )nn 1n

a nP n 1 Q n 1 P n 2 Q2 2

= +

( )na P n 1 Q = +

n n 1d a a = ( ) ( )d P n 1 Q P n 2 Q = +

d Q = Ans (d).


47/63

Cass =ercise 84.

_I0038. /f 12th

term of an A.P. is 1" and thesum of the first four terms is 24!$hat is the sum of first 10 terms6


48/63

Cass =ercise 84.

_I003

8. /f 12thterm of an A.P. is 1" and the sumof the first four terms is 24! $hat is the

sum of first 10 terms6

A. 'i%en that!

a125 a1311d 5 1" . . . (i)

S45 2(2a13"d) 5 24 . . . (ii)

So%ing (i) and (ii) simutaneous! $e get!

a15 ! d 5 2

S105 &(2a13d) 5 &(1>1>) 5 0


49/63

Cass =ercise 8&.

_I0048. Find the %aue of n so that#e an A.M. #et$een a and # (a! #are ositi%e).

+ ++

+

n 1 n 1

n n

a b

a b


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Cass =ercise 8&.

A. 'i%en that!

_I004

8. Find the %aue of n so that

#e an A.M. #et$een a and # (a! # areositi%e).

+ ++

+

n 1 n 1

n n

a b

a b

n 1 n 1

n n

a b a b

2a b

+ ++ +

=+n 1 n 1 n na b a b ab+ + + = +

,i%iding throughout # #n31! $e get!n 1 n

a a a1

b b b

+ + = + n

a a a1 1

b b b

=


51/63

Cass =ercise 8&.

_I004

8. Find the %aue of n so that

#e an A.M. #et$een a and # (a! # areositi%e).

+ ++

+

n 1 n 1

n n

a b

a b

na a a

1 a b 1 1 0b b b

=

Q

na a a

1 1b b b

=

n 5 0


52/63

Cass =ercise 8.

_I0048. &" A.M.s are inserted #et$een 2 and>. Find the 2EthA.M.


53/63

Cass =ercise 8.

_I004

8. &" A.M.s are inserted #et$een 2 and >.Find the 2EthA.M.

A. Common difference

b a 98 2 48

n 1 53 1 27

= = =

+ +

27

48A a 27d 2 27 50

27

= + = + =


54/63

Cass =ercise 8E.

_I0058. /f the "rd

term of a '.P. is 4! $hat isthe roduct of the first fi%e terms6


55/63

Cass =ercise 8E.

_I005

8. /f the "rdterm of a '.P. is 4! $hat is theroduct of the first fi%e terms6

A. 9et the first & terms of the '.P. #e

2

2

a a, , a, a, a

equired roduct 5 a&

5 (a")&

54&


56/63

Cass =ercise 8>.

_I0058. /f the 4th

! E

th

! 10

th

term of a '.P. are! q! r resecti%e! then

(a) 25 q23r2 (#) q25 r

(c) 25 qr (d) qr3q3q 5 0

(M.-.. 1&)


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Cass =ercise 8>.

_I005

8. /f the 4th! Eth! 10thterm of a '.P. are ! q!r resecti%e! then

(a) 25 q23r2 (#) q25 r

(c) 25 qr(d) qr3q3q 5 0

(M.-.. 1&)

A. 9et the first term of the '.P. #e

and common ratio #e .

Ans (#)

5 "! q 5 ! r 5

-o$! r 5 212

5 ()2

5 q2


58/63

Cass =ercise 8.

_I0068. Find the sum to n terms of thesequence ! ! ! . . .


59/63

Cass =ercise 8.

_I006

8. Find the sum to n terms of the sequence! ! ! . . .

A. Sn5 (333 . . .Kn termsL)

[ ]( )n6

S 9 99 999 . . . n te!"9

= + + +

( ) ( ) ( ) [ ]{ }n6

S 10 1 100 1 1000 1 . . . n te!"9 = + + +

[ ]( ){ }2 3n6

S 10 10 10 . . . n te!" n9

= + +

( )( )

=

n

n

10 16S 10 n9 10 1

( )nn6 10

S 10 1 n9 9

=


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Cass =ercise 810.

_I006

8. o$ man terms of the '.P.

are needed to gi%e the sum 6

2 1 1 3, , , , . . .

9 3 2 4

5572

A. Common ratio

1

33

2 2

9

= =

9et the required num#er of terms #e n.

n

n

n

3

155 2 2 2 32S 1

372 9 9 5 21

2

= = =


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Cass =ercise 810.

_I006

8. o$ man terms of the '.P.

are needed to gi%e the sum 6

2 1 1 3, , , , . . .

9 3 2 4

5572

n55 2 2 3

172 9 5 2

= n

3 55 9 5 24312 72 2 2 32 = + =

n 53 3

2 2

=

n 5 &


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04. Sequence and Series-1

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