Progressions
1. Arithmetic Progression
2. Geometric progression
3. Harmonic Progression
and other series
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1. Arithmetic progression : In arithmetic progression, each term is greater than the last number by a certain fixed number.
Representation of Arithmetic progressions
Lets say, a series start with the number a and the fixed number by which it term increases in arithmetic progression is d, then the series will be,
a, a + d, a + 2d , a + 3d, a + 4d, a + 5d, a + 6d, ...
General Term of arithmetic series: The general term of an arithmetic series is represented by
Tn = a + (n -1)d
Sum of n terms of arithmetic series :
Sn = (n/2){ 2a + (n-1)d}
or Sn = (n/2)(first term + last term)
Derivation of sum of arithmetic series:
Lets say we have to find the sum of the n term of the series, beginning with the term a and with the common difference d,
then,
Sum = a + (a + d) + (a + 2d) + (a +3d) + ... + {a + (n-1)d}
Now, Since there are n terms in the series
then,
Sum = na + d + 2d +3d + ... + (n-1)d
= na + d{ 1 + 2+ 3 + (n-1)}
= na + d{n(n-1)}/2
= n[ a + {(n-1)d}/2]
= (n/2){ 2a + (n-1)d}
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2. Geometric progression : In geometric progression, ratio of 'n' term and the (n-1) term is always constant.
General representation of the series is :
a , ar, ar2, ar3, ar4, ar5, ....arn
General term of the is :
Tn = arn
Sum of n terms of arithmetic series :
If the common ratio 'r' is greater than 1 (r >1)
then,
Sn = a(rn - 1) / ( r -1)
else,If the common ratio 'r' is less than 1 (r <1)
Sn = a(rn - 1) / ( r -1)
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3. Harmonic progression : A series a, b,c is said to be in harmonic progression if
(a/c) = (a - b)/(b - c)
Also , the reciprocals of all numbers of a series are in arithmetic progression, then these numbers are said to be in harmonic progressions.
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rest of the series will covered by through solved questions
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Solved Questions:
1. Find the odd man out in the following series
(i). 1, 4, 9, 16, 23, 25, 36
(ii). 2, 5, 10, 50, 500, 5000
(iii) 10, 25, 45, 54, 60, 75, 80
(iv) 1, 8, 27, 64, 124, 216, 343
Solution:
(i). 1, 4, 9, 16, 23, 25, 36
Answer: All the numbers here are the squares of natural starting from number 1 except for
23 that does not belong here.
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(ii). 2, 5, 10, 50, 500, 5000
Answer: third term is multiplication of 2nd and 3rd term( 10 = 2 x 5), and 4th term is the
multiplication of 4th and 5th term ( 50 = 5 x 10) but the last term 5000 does not
belong here as it not the product of the preivous terms ( should have been 500 x 50 =
25000)
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(iii) 10, 25, 30, 54, 60, 75, 80
Answer: 54 does not belong in this series. 15 is added to each term in the series to obtain
next term but not 54.
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(iv) 1, 8, 27, 64, 124, 216, 343
Answer: Each number here is the cube of natural number starting with 1. but the 124 does not
belong here as it is not the cube of number 5.
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2. Find the sum of first 20 natural numbers
Solution: From the formula defined above for sum of AP,
we have,
S20 = (20/2) { 2 + 19}
= (10) ( 21)
= 210
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3. There is an arithmetic progression that start at 1, 3, 5 ... then the number 55 will come at the
term
(1) 27th (2) 26th
(3) 25th (4) 28th
(5) 29th
Solution: From the formula defined above the nth term is defined as
T(n) = a + (n-1)d
here, a = 1 and d =2 and Put T =55, we need to find n
putting the value,
55 = 1 + (n -1 )2
=> 55 = 1 + 2n - 2
=> 55 + 2 -1 = 2n
=> 56 = 2n
=> n = 28
Therefore, 28th term will be equal to 55. Hence option (4) is correct.
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1. Arithmetic Progression
2. Geometric progression
3. Harmonic Progression
and other series
___________________________________________________________
1. Arithmetic progression : In arithmetic progression, each term is greater than the last number by a certain fixed number.
Representation of Arithmetic progressions
Lets say, a series start with the number a and the fixed number by which it term increases in arithmetic progression is d, then the series will be,
a, a + d, a + 2d , a + 3d, a + 4d, a + 5d, a + 6d, ...
General Term of arithmetic series: The general term of an arithmetic series is represented by
Tn = a + (n -1)d
Sum of n terms of arithmetic series :
Sn = (n/2){ 2a + (n-1)d}
or Sn = (n/2)(first term + last term)
Derivation of sum of arithmetic series:
Lets say we have to find the sum of the n term of the series, beginning with the term a and with the common difference d,
then,
Sum = a + (a + d) + (a + 2d) + (a +3d) + ... + {a + (n-1)d}
Now, Since there are n terms in the series
then,
Sum = na + d + 2d +3d + ... + (n-1)d
= na + d{ 1 + 2+ 3 + (n-1)}
= na + d{n(n-1)}/2
= n[ a + {(n-1)d}/2]
= (n/2){ 2a + (n-1)d}
___________________________________________________________
2. Geometric progression : In geometric progression, ratio of 'n' term and the (n-1) term is always constant.
General representation of the series is :
a , ar, ar2, ar3, ar4, ar5, ....arn
General term of the is :
Tn = arn
Sum of n terms of arithmetic series :
If the common ratio 'r' is greater than 1 (r >1)
then,
Sn = a(rn - 1) / ( r -1)
else,If the common ratio 'r' is less than 1 (r <1)
Sn = a(rn - 1) / ( r -1)
___________________________________________________________
3. Harmonic progression : A series a, b,c is said to be in harmonic progression if
(a/c) = (a - b)/(b - c)
Also , the reciprocals of all numbers of a series are in arithmetic progression, then these numbers are said to be in harmonic progressions.
__________________________________________________________
rest of the series will covered by through solved questions
__________________________________________________________
Solved Questions:
1. Find the odd man out in the following series
(i). 1, 4, 9, 16, 23, 25, 36
(ii). 2, 5, 10, 50, 500, 5000
(iii) 10, 25, 45, 54, 60, 75, 80
(iv) 1, 8, 27, 64, 124, 216, 343
Solution:
(i). 1, 4, 9, 16, 23, 25, 36
Answer: All the numbers here are the squares of natural starting from number 1 except for
23 that does not belong here.
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(ii). 2, 5, 10, 50, 500, 5000
Answer: third term is multiplication of 2nd and 3rd term( 10 = 2 x 5), and 4th term is the
multiplication of 4th and 5th term ( 50 = 5 x 10) but the last term 5000 does not
belong here as it not the product of the preivous terms ( should have been 500 x 50 =
25000)
__________________________________________________________
(iii) 10, 25, 30, 54, 60, 75, 80
Answer: 54 does not belong in this series. 15 is added to each term in the series to obtain
next term but not 54.
__________________________________________________________
(iv) 1, 8, 27, 64, 124, 216, 343
Answer: Each number here is the cube of natural number starting with 1. but the 124 does not
belong here as it is not the cube of number 5.
__________________________________________________________
2. Find the sum of first 20 natural numbers
Solution: From the formula defined above for sum of AP,
we have,
S20 = (20/2) { 2 + 19}
= (10) ( 21)
= 210
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3. There is an arithmetic progression that start at 1, 3, 5 ... then the number 55 will come at the
term
(1) 27th (2) 26th
(3) 25th (4) 28th
(5) 29th
Solution: From the formula defined above the nth term is defined as
T(n) = a + (n-1)d
here, a = 1 and d =2 and Put T =55, we need to find n
putting the value,
55 = 1 + (n -1 )2
=> 55 = 1 + 2n - 2
=> 55 + 2 -1 = 2n
=> 56 = 2n
=> n = 28
Therefore, 28th term will be equal to 55. Hence option (4) is correct.
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