Mean and progression in mathmatics
Arithmetic Progression
An arithmetic progression is a sequence of numbers in which each term is derived from the preceding term by adding or subtracting a fixed number called the common difference "d"
For example, the sequence 9, 6, 3, 0,-3, .... is an arithmetic progression with -3 as the common difference. The progression -3, 0, 3, 6, 9 is an Arithmetic Progression (AP) with 3 as the common difference.
The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d and so on.
Thus nth term of an AP series is T = a + (n - 1) d, where T = n term and a = first term. Here d = common difference = T (n+1)- T (n).
Sum of first n terms of an AP: S =(n/2)[2a + (n- 1)d]
The sum of n terms is also equal to the formula = n/2(a+l)
where l is the last term.
T(n+1) = S(n+1) - S(n) , where T = n+1 term
When three quantities are in AP, the middle one is called as the arithmetic mean of the other two. If a, b and c are three terms in AP then b = (a+c)/2
Geometric Progression
A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. For example, the sequence 4, -2, 1, - 1/2,.... is a Geometric Progression (GP) for which - 1/2 is the common ratio.
The general form of a GP is a, ar, ar² , ar³ and so on.
The nth term of a GP series is T = arⁿ⁻¹ , where a = first term and r = common ratio = Tₙ₊₁ /T ₙ) .
The formula applied to calculate sum of first n terms of a GP=a(rⁿ-1)/r-1
When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b =√ac.
The sum of infinite terms of a GP series S = a/(1-r) where 0< r<1.
If a is the first term, r is the common ratio of a finite G.P. consisting of x terms, then the nth term from the end will be = arˣ⁻ⁿ.
Harmonic Progression
A series of terms is known as a HP series when their reciprocals are in arithmetic progression.
Example: 1/a, 1/(a+d), 1/(a+2d), and so on are in HP because a, a + d, a + 2d are in AP.
The n term of a HP series is T =1/ [a + (n -1) d].
In order to solve a problem on Harmonic Progression, one should make the corresponding AP series and then solve the problem.
nth term of H.P. = 1/(nth term of corresponding A.P.)
If three terms a, b, c are in HP, then b =2ac/(a+c).
Arithmatic mean
It is the average of all the numbers in a series.
It is given by = sum of all the numbers/total numbers
Mean = ( n1+n2+n3+.....nx)/x
Geometric mean
The geometric mean is defined as the n th root of the product of n numbers, i.e., for a set of numbers x 1 , x 2 , ..., x n , the geometric mean is defined as
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, 4.
Let there are two numbers ‘a’ and ‘b’, a, b > 0
then AM = a+b/2
GM =√ab
HM =2ab/a+b
∴ AM × HM =a+b/2 × 2ab/a+b = ab = (√ab)2 = (GM)2
Note that these means are in G.P.
Hence AM.GM.HM follows the rules of G.P.
i.e. G.M. =√A.M. × H.M.
Now, let us see the difference between AM and GM
AM – GM =a+b/2 – √ab
=(√a )+(√b)–2√a√b/2
i.e. AM > GM
Similarly,
G.M. – H.M. = √ab –2ab/a+b
=√ab/a+b (√a – √b)2 > 0
So. GM > HM
Combining both results, we get
AM > GM > HM
Arithmetic Progression
An arithmetic progression is a sequence of numbers in which each term is derived from the preceding term by adding or subtracting a fixed number called the common difference "d"
For example, the sequence 9, 6, 3, 0,-3, .... is an arithmetic progression with -3 as the common difference. The progression -3, 0, 3, 6, 9 is an Arithmetic Progression (AP) with 3 as the common difference.
The general form of an Arithmetic Progression is a, a + d, a + 2d, a + 3d and so on.
Thus nth term of an AP series is T = a + (n - 1) d, where T = n term and a = first term. Here d = common difference = T (n+1)- T (n).
Sum of first n terms of an AP: S =(n/2)[2a + (n- 1)d]
The sum of n terms is also equal to the formula = n/2(a+l)
where l is the last term.
T(n+1) = S(n+1) - S(n) , where T = n+1 term
When three quantities are in AP, the middle one is called as the arithmetic mean of the other two. If a, b and c are three terms in AP then b = (a+c)/2
Geometric Progression
A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. For example, the sequence 4, -2, 1, - 1/2,.... is a Geometric Progression (GP) for which - 1/2 is the common ratio.
The general form of a GP is a, ar, ar² , ar³ and so on.
The nth term of a GP series is T = arⁿ⁻¹ , where a = first term and r = common ratio = Tₙ₊₁ /T ₙ) .
The formula applied to calculate sum of first n terms of a GP=a(rⁿ-1)/r-1
When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e. b =√ac.
The sum of infinite terms of a GP series S = a/(1-r) where 0< r<1.
If a is the first term, r is the common ratio of a finite G.P. consisting of x terms, then the nth term from the end will be = arˣ⁻ⁿ.
Harmonic Progression
A series of terms is known as a HP series when their reciprocals are in arithmetic progression.
Example: 1/a, 1/(a+d), 1/(a+2d), and so on are in HP because a, a + d, a + 2d are in AP.
The n term of a HP series is T =1/ [a + (n -1) d].
In order to solve a problem on Harmonic Progression, one should make the corresponding AP series and then solve the problem.
nth term of H.P. = 1/(nth term of corresponding A.P.)
If three terms a, b, c are in HP, then b =2ac/(a+c).
Arithmatic mean
It is the average of all the numbers in a series.
It is given by = sum of all the numbers/total numbers
Mean = ( n1+n2+n3+.....nx)/x
Geometric mean
The geometric mean is defined as the n th root of the product of n numbers, i.e., for a set of numbers x 1 , x 2 , ..., x n , the geometric mean is defined as
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, 4.
Let there are two numbers ‘a’ and ‘b’, a, b > 0
then AM = a+b/2
GM =√ab
HM =2ab/a+b
∴ AM × HM =a+b/2 × 2ab/a+b = ab = (√ab)2 = (GM)2
Note that these means are in G.P.
Hence AM.GM.HM follows the rules of G.P.
i.e. G.M. =√A.M. × H.M.
Now, let us see the difference between AM and GM
AM – GM =a+b/2 – √ab
=(√a )+(√b)–2√a√b/2
i.e. AM > GM
Similarly,
G.M. – H.M. = √ab –2ab/a+b
=√ab/a+b (√a – √b)2 > 0
So. GM > HM
Combining both results, we get
AM > GM > HM
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