Arithmetic, geometric and harmonic progression
tkjainbkn
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May 28, 2010
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5 DECEMBER 09 www.afterschool.tk 1
ARITHMETIC, GEOMETRIC AND
HARMONIC PROGRESSION
by :
DR. T.K. JAIN
AFTERSCHO☺OL
centre for social entrepreneurship
sivakamu veterinary hospital road
bikaner 334001 rajasthan, india
FOR – PGPSE / CSE PARTICIPANTS
[email protected]
mobile : 91+9414430763
ARITHMETIC, GEOMETRIC AND
HARMONIC PROGRESSION
by :
DR. T.K. JAIN
AFTERSCHO☺OL
centre for social entrepreneurship
sivakamu veterinary hospital road
bikaner 334001 rajasthan, india
FOR – PGPSE / CSE PARTICIPANTS
[email protected]
mobile : 91+9414430763
5 DECEMBER 09 www.afterschool.tk 2
My words.....
My purpose here is to give an introduction of arithmetic,
geometric and harmonic progression . I welcome your
suggestions. I also request you to help me in spreading
social entrepreneurship across the globe – for which I
need support of you people – not of any VIP. With your
help, I can spread the ideas – for which we stand....
My words.....
My purpose here is to give an introduction of arithmetic,
geometric and harmonic progression . I welcome your
suggestions. I also request you to help me in spreading
social entrepreneurship across the globe – for which I
need support of you people – not of any VIP. With your
help, I can spread the ideas – for which we stand....
5 DECEMBER 09 www.afterschool.tk 3
What is ARITHMETIC
PROGRESSION ?
3 tests of AP :
1 there must be a fixed interval between any
two digits in a seris
2. there must not be any other reason for the
numbers to come in that order
3 the numbers either increase or decrease
What is ARITHMETIC
PROGRESSION ?
3 tests of AP :
1 there must be a fixed interval between any
two digits in a seris
2. there must not be any other reason for the
numbers to come in that order
3 the numbers either increase or decrease
5 DECEMBER 09 www.afterschool.tk 4
Example of AP ?
8,13,18, 23,28,33 ....
Here we have an AP, because there is a fixed
gap and the series is increasing
in a series we have 3 terms :
a = first term = 8
d = difference between any two terms = 5
n = number of terms
Example of AP ?
8,13,18, 23,28,33 ....
Here we have an AP, because there is a fixed
gap and the series is increasing
in a series we have 3 terms :
a = first term = 8
d = difference between any two terms = 5
n = number of terms
5 DECEMBER 09 www.afterschool.tk 5
Basic formula of AP ?
In order to find sum of some terms, you can
use any of these formula :
1. n/2 ( 2a+(n-1)d)
2 n/2 (first term + last term)
Basic formula of AP ?
In order to find sum of some terms, you can
use any of these formula :
1. n/2 ( 2a+(n-1)d)
2 n/2 (first term + last term)
5 DECEMBER 09 www.afterschool.tk 6
Example of the formula :
What will be the sum of first 100 terms of this
series :
8,13,18,23,28 ...
n/2 ( 2a+(n-1)d)
n = 100, a = 8, d = 5
100/2 (2*8 + (100-1)*5)
=50(16 + 495)
=25550 answer
Example of the formula :
What will be the sum of first 100 terms of this
series :
8,13,18,23,28 ...
n/2 ( 2a+(n-1)d)
n = 100, a = 8, d = 5
100/2 (2*8 + (100-1)*5)
=50(16 + 495)
=25550 answer
5 DECEMBER 09 www.afterschool.tk 7
How to identify a term in an AP ?
In order to identify any term, use this formula :
Tn = a + (n-1)d
what is 30
th
term in this series :
8,13,18,23,28 ...
= 8+(30-1)5
=8+145
=153 ans
How to identify a term in an AP ?
In order to identify any term, use this formula :
Tn = a + (n-1)d
what is 30
th
term in this series :
8,13,18,23,28 ...
= 8+(30-1)5
=8+145
=153 ans
5 DECEMBER 09 www.afterschool.tk 8
What wil be 17
th
term in the series
: 103, 123, 143, 163 .... ?
Formula tn= a + (n-1) d
where, a = starting term = 103
d = difference (123- 103 = 20) , n = number of
term
=103 + (17-1) * 20
=103 + 320 = 423 answer
What wil be 17
th
term in the series
: 103, 123, 143, 163 .... ?
Formula tn= a + (n-1) d
where, a = starting term = 103
d = difference (123- 103 = 20) , n = number of
term
=103 + (17-1) * 20
=103 + 320 = 423 answer
5 DECEMBER 09 www.afterschool.tk 9
There are 31 terms in an AP . The first term is
100340 and the last term is 100580. What is the sum
of that series ?
Formula =n/2 * (first term + last term)
n = 31, first term = 100340, last term =100580
=31/2 *(100340+100580)
=3114260 answer
There are 31 terms in an AP . The first term is
100340 and the last term is 100580. What is the sum
of that series ?
Formula =n/2 * (first term + last term)
n = 31, first term = 100340, last term =100580
=31/2 *(100340+100580)
=3114260 answer
5 DECEMBER 09 www.afterschool.tk 10
What is GP ? (geometric
progression)
When a series is rising at an increasing fast
speed, it is geometric progression
example :
2,4,8,16,32,64 .....here a= 2 (starting term)
raio = 16/8 = 2 (you make pick up any two
terms, the ratio will be the same)
What is GP ? (geometric
progression)
When a series is rising at an increasing fast
speed, it is geometric progression
example :
2,4,8,16,32,64 .....here a= 2 (starting term)
raio = 16/8 = 2 (you make pick up any two
terms, the ratio will be the same)
5 DECEMBER 09 www.afterschool.tk 11
Essential components of GP?
Take any GP : Let us say : 3,9,27, 81,243
There are 3 terms :
1. starting term = a = 3
2. growth as ratio of two terms : r = 27/9 =3
3.n = number of terms = n=5
Essential components of GP?
Take any GP : Let us say : 3,9,27, 81,243
There are 3 terms :
1. starting term = a = 3
2. growth as ratio of two terms : r = 27/9 =3
3.n = number of terms = n=5
5 DECEMBER 09 www.afterschool.tk 12
Can ratio be less than 1 ?
Yes,
in that case the series will be a decreasing
series
example :
81,27,9,3,1,1/3, 1/9,1/27 ....
Can ratio be less than 1 ?
Yes,
in that case the series will be a decreasing
series
example :
81,27,9,3,1,1/3, 1/9,1/27 ....
5 DECEMBER 09 www.afterschool.tk 13
What is the sum of this series for
5 terms:
1^2, 2^2, 3^2 . . . .
Formula :( n (n+1) (2n+1) )/ 6
=(5(6)(11))/6
=55 answer
What is the sum of this series for
5 terms:
1^2, 2^2, 3^2 . . . .
Formula :( n (n+1) (2n+1) )/ 6
=(5(6)(11))/6
=55 answer
5 DECEMBER 09 www.afterschool.tk 14
What is the sum of this series for
5 terms:
1^3, 2^3, 3^3 . . . .
Formula = ((n(n+1)/2)^2)
(5(5+1)/2)^2
= 225 answer
What is the sum of this series for
5 terms:
1^3, 2^3, 3^3 . . . .
Formula = ((n(n+1)/2)^2)
(5(5+1)/2)^2
= 225 answer
5 DECEMBER 09 www.afterschool.tk 15
What is the sum of this infinite
series : 512, 256,128,64......for
first 10 terms ???
Formula = a / (1-r)
a=512, r = 256/512 = ½ = .5
= 512/ (1-.5)
= 1024 ans.
What is the sum of this infinite
series : 512, 256,128,64......for
first 10 terms ???
Formula = a / (1-r)
a=512, r = 256/512 = ½ = .5
= 512/ (1-.5)
= 1024 ans.
5 DECEMBER 09 www.afterschool.tk 16
What is sum of this series :
2,4,8,16 .for first 30 terms ???
'Sn = a (r^n -1) / (r-1)
a= starting term r = ratio n = 30
=2(2^30 – 1) / (2-1)
=2 * 1073741823
=2147483646 answer
What is sum of this series :
2,4,8,16 .for first 30 terms ???
'Sn = a (r^n -1) / (r-1)
a= starting term r = ratio n = 30
=2(2^30 – 1) / (2-1)
=2 * 1073741823
=2147483646 answer
5 DECEMBER 09 www.afterschool.tk 17
What is the sum of first 8 terms of
this series : 8, 64, 512 .....
This is GP
formula for sum is :
a (r^n – 1) / (r-1)
a = first term (8), r = ratio = 64/8 = 8
8 * (8^8 - 1) / (8-1)
=19173960 answer
What is the sum of first 8 terms of
this series : 8, 64, 512 .....
This is GP
formula for sum is :
a (r^n – 1) / (r-1)
a = first term (8), r = ratio = 64/8 = 8
8 * (8^8 - 1) / (8-1)
=19173960 answer
5 DECEMBER 09 www.afterschool.tk 18
What is the sum of first 5 terms of
this series :
81,27,9 ...
This is GP
formula for sum = a (1-r^n) / (1-r)
a = 81, r =27/81 = 1/3, n =5
= 81 (1-(1/3)^5) / (1-(1/3))
=121 answer
What is the sum of first 5 terms of
this series :
81,27,9 ...
This is GP
formula for sum = a (1-r^n) / (1-r)
a = 81, r =27/81 = 1/3, n =5
= 81 (1-(1/3)^5) / (1-(1/3))
=121 answer
5 DECEMBER 09 www.afterschool.tk 19
Which of these is geomtric
progression ?
3,5,88,122
3,5,7,9,11...
4,16,256 ...
only the last one (4,16,256....) is a GP
Which of these is geomtric
progression ?
3,5,88,122
3,5,7,9,11...
4,16,256 ...
only the last one (4,16,256....) is a GP
5 DECEMBER 09 www.afterschool.tk 20
In order to study at ISB Hyderabad, Mayank starts
saving money. First day he saves Rs. 200000. Next day
he saves half of this money. This process continues.
Now this becomes his habit. If Mayank continues
saving like this, how much money can he save in his
lifetime?
Formula = a / (1-r)
a = first term (2,00,000) , r = ratio (½)
=200000/.5
= 4 lakhs only answer
In order to study at ISB Hyderabad, Mayank starts
saving money. First day he saves Rs. 200000. Next day
he saves half of this money. This process continues.
Now this becomes his habit. If Mayank continues
saving like this, how much money can he save in his
lifetime?
Formula = a / (1-r)
a = first term (2,00,000) , r = ratio (½)
=200000/.5
= 4 lakhs only answer
5 DECEMBER 09 www.afterschool.tk 21
If after 1 year, Mayank can get Rs. 50 lakhs
in one shot. How much is it worth now. Rate
of interest is 20% ?
Formula of present value
=( amount*100) / (100 + (rate * time) )
=(5000000*100)/(120)
=4166666 answer
If after 1 year, Mayank can get Rs. 50 lakhs
in one shot. How much is it worth now. Rate
of interest is 20% ?
Formula of present value
=( amount*100) / (100 + (rate * time) )
=(5000000*100)/(120)
=4166666 answer
5 DECEMBER 09 www.afterschool.tk 22
What is the worth of Rs. 100000 now after 10
years at 20% annual compounding interest ?
Formula = amount (1+rate ) ^ n
n = number of years
100000 * (1+.2)^10
=619174 answer
What is the worth of Rs. 100000 now after 10
years at 20% annual compounding interest ?
Formula = amount (1+rate ) ^ n
n = number of years
100000 * (1+.2)^10
=619174 answer
5 DECEMBER 09 www.afterschool.tk 23
A goes to his school @ 20 km per
hour and returns back to home @
30 km per hour. What is average
speed ?
Average of 20 and 30 is 25, but here we have to
calculate harmonic mean.
Formula of harmonic mean
=reverse of 1/n ( 1/x + 1/y)
or : (2XY / (X+Y))
=1/2 (1/20 + 1/30) = ½ * 50 / 600
= 24 answer
A goes to his school @ 20 km per
hour and returns back to home @
30 km per hour. What is average
speed ?
Average of 20 and 30 is 25, but here we have to
calculate harmonic mean.
Formula of harmonic mean
=reverse of 1/n ( 1/x + 1/y)
or : (2XY / (X+Y))
=1/2 (1/20 + 1/30) = ½ * 50 / 600
= 24 answer
5 DECEMBER 09 www.afterschool.tk 24
What is harmonic progression ?
It is just reverse of AP
for example : 2,4,6,8 is an AP
HP = 1/2, 1/4, 1/6 ,1/8
harmonic progression will convert into AP
when you reverse it.
What is harmonic progression ?
It is just reverse of AP
for example : 2,4,6,8 is an AP
HP = 1/2, 1/4, 1/6 ,1/8
harmonic progression will convert into AP
when you reverse it.
5 DECEMBER 09 www.afterschool.tk 25
Find AM,GM,HM between 20
and 30 ?
AM= (20+30)/2 = 25
GM = sqrt (20*30) = 24.49
HM = REVERSE OF ½ ( 1/20+1/30)=24
Find AM,GM,HM between 20
and 30 ?
AM= (20+30)/2 = 25
GM = sqrt (20*30) = 24.49
HM = REVERSE OF ½ ( 1/20+1/30)=24
5 DECEMBER 09 www.afterschool.tk 26
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5 DECEMBER 09 www.afterschool.tk 28
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5 DECEMBER 09 www.afterschool.tk 32
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5 DECEMBER 09 www.afterschool.tk 33
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5 DECEMBER 09 www.afterschool.tk 34
Download links for material in english
http://www.authorstream.com/presentation/tkjainbkn-146799-english-error-spotting-sentence-im-law-cat-gmat-mba-management-business-research-cfp-cfa-frm-cpa-ca-cs-icwa-india-rajasthan-improvement-entertainment-ppt-powerpoint/
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http://www.scribd.com/doc/14660441/English-Afterschoool-23-May
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Download links for material in english
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http://www.docstoc.com/docs/3921499/ENGLISH-%E2%80%93-ERROR-SPOTTING-AND-SENTENCE-IMPROVEMENT
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http://www.scribd.com/doc/19641980/Error-Spotting
http://www.scribd.com/doc/11629005/English-Error-Spotting-and-Sentence-Improvement
http://www.scribd.com/doc/14660441/English-Afterschoool-23-May
http://www.scribd.com/doc/6583519/English-Afterschoool-21-May
http://www.scribd.com/doc/6583520/English-Afterschoool-21-May-2
5 DECEMBER 09 www.afterschool.tk 35
Download links for material on
English
http://www.scribd.com/doc/6583315/English-Improvement-Afterschoool
http://www.scribd.com/doc/6583518/English-20-May-Afterschoool
http://www.scribd.com/doc/28531795/Mock-Paper-Cat-Rmat-Mat-Sbi-Bank-Po-Aptitude-Tests
Download links for material on
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http://www.scribd.com/doc/6583315/English-Improvement-Afterschoool
http://www.scribd.com/doc/6583518/English-20-May-Afterschoool
http://www.scribd.com/doc/28531795/Mock-Paper-Cat-Rmat-Mat-Sbi-Bank-Po-Aptitude-Tests
5 DECEMBER 09 www.afterschool.tk 36
Links
http://www.scribd.com/doc/14647398/English-Improvement-and-Word-Power
http://www.scribd.com/doc/14660508/General-Knowledge-24-May-3
http://www.scribd.com/doc/19628963/Mathematics
http://www.scribd.com/doc/19492878/Direct-and-Indirect-Speech-in-English
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http://www.scribd.com/doc/14660508/General-Knowledge-24-May-3
http://www.scribd.com/doc/19628963/Mathematics
http://www.scribd.com/doc/19492878/Direct-and-Indirect-Speech-in-English
5 DECEMBER 09 www.afterschool.tk 37
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5 DECEMBER 09 www.afterschool.tk 38
ADDITIONAL LINKS
http://www.scribd.com/doc/11692763/Advanced-Mathematics-for-GMAT-CAT-MAT
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http://www.scribd.com/doc/11625819/Basic-Mathematics-for-Cat-GMAT-Mat
http://www.scribd.com/doc/23407954/mathematics-for-OPENMAT-MAT-CAT-GMAT-25-april
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http://www.scribd.com/doc/6583520/English-Afterschoool-21-May-2
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http://www.scribd.com/doc/23407945/MATHEMATICS-FOR-ATMA-CAT-MAT-GMAT-BANK-PO-GRE
ADDITIONAL LINKS
http://www.scribd.com/doc/11692763/Advanced-Mathematics-for-GMAT-CAT-MAT
http://www.scribd.com/doc/23407929/Advanced-Mathematics-for-GMAT-CAT-MAT
http://www.scribd.com/doc/11625819/Basic-Mathematics-for-Cat-GMAT-Mat
http://www.scribd.com/doc/23407954/mathematics-for-OPENMAT-MAT-CAT-GMAT-25-april
http://www.scribd.com/doc/23407934/Advanced-Mathematics-for-GMAT-CAT-MAT2
http://www.scribd.com/doc/6583520/English-Afterschoool-21-May-2
http://www.scribd.com/doc/23300964/Advanced-Mathematics-for-GMAT-CAT-MAT2
http://www.scribd.com/doc/23407945/MATHEMATICS-FOR-ATMA-CAT-MAT-GMAT-BANK-PO-GRE
5 DECEMBER 09 www.afterschool.tk 39
http://www.scribd.com/doc/6583303/Reasoning-Afterschoool
http://www.scribd.com/doc/6583303/Reasoning-Afterschoool
5 DECEMBER 09 www.afterschool.tk 40
http://www.scribd.com/doc/23393680/REASONING-AFTERSCHOOOL
http://www.scribd.com/doc/23393719/reasonning-44
http://www.scribd.com/doc/6583347/DI-and-Reasoning
http://www.scribd.com/doc/23407636/10-July-reasoning
http://www.scribd.com/doc/6583273/Reasoning-Quiz
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http://www.scribd.com/doc/14705025/17-Reasoning-II
http://www.scribd.com/doc/23393478/10-July-reasoning-II
http://www.scribd.com/doc/23393680/REASONING-AFTERSCHOOOL
http://www.scribd.com/doc/23393719/reasonning-44
http://www.scribd.com/doc/6583347/DI-and-Reasoning
http://www.scribd.com/doc/23407636/10-July-reasoning
http://www.scribd.com/doc/6583273/Reasoning-Quiz
http://www.scribd.com/doc/23393476/10-July-reasoning
http://www.scribd.com/doc/23393716/REASONING-QUIZ
http://www.scribd.com/doc/14705025/17-Reasoning-II
http://www.scribd.com/doc/23393478/10-July-reasoning-II
5 DECEMBER 09 www.afterschool.tk 41
Free download useful material ...
http://www.scribd.com/doc/23393316/general-knowledge
http://www.scribd.com/doc/23609752/Group-Discussion-Afterschoool
http://www.scribd.com/doc/6583547/General-Knowledge-24-May
Free download useful material ...
http://www.scribd.com/doc/23393316/general-knowledge
http://www.scribd.com/doc/23609752/Group-Discussion-Afterschoool
http://www.scribd.com/doc/6583547/General-Knowledge-24-May
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