Progression In Math

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Sub Topics Geometric Progression Arithmetic Progression Arithmetic Progression Formula Harmonic Progression

Progression in maths can be defined as sequence of terms that increases in a particular pattern. There are different types of progression in Math which are mentioned below: Arithmetic progression Geometric progression Harmonic progression. Let's have small introduction about all progression. First we will see Arithmetic Progression. Arithmetic progression can be defined as a sequence of number in which we get difference between two successive Numbers which is always constant. Representation of an arithmetic progression is a, a + d, a + 2d, a + 3d, a + 4d …...... a + (n – 1) d, here value of 'a' denotes initial value and 'd' denotes common difference. For example: 1, 2, 3, 4, 5, 6 here initial value of A.P is 1 and difference between two number is also 1. Formula for arithmetic progression is given below: If we find n th term of A.P. then formula is given by = a + (n – 1)d. If we find sum of 'n' terms in A.P. then formula is given by = n / 2 (2a + (n – 1) d). Geometric progression: - Geometric progression can be defined as sequence of number in which we get Ratio of two successive numbers which is a constant. Representation of Geometric Progression’s’ is given by: a, ar, ar2, ar3......., ar(n – 1). Here initial number is given as 'a' and common ratio is given as 'r'. For example: 1, 2, 4, 8, 16. here the initial number is 1 and common ratio is 2. Harmonic progression: - Harmonic progression can be defined as sequence of number in which we get difference of successive denominator of Fractions which is equal. Representation of harmonic Functions is given as: a, a/d, a/(a + d), a/(a + 2d),..... a/a + nd). This is all about progression in maths.

Progression is defined as progress in any field or in a particular format and pattern. It refers to any ongoing sequence which means a progress or an increment. Term Geometric Progression refers to a sequence in which successor term of each term is obtained by multiplying a constant term. This constant term is multiplied by every term to obtain the next term. This constant term is called as Common Ratio and it is represented by ‘r’. Basically, progression is classified in three types that are Arithmetic Progression, geometric progression and Harmonic Progression. Arithmetic progression is defined as a series in which proceeding or successor term is determined by adding a common difference, this common difference is constant for a series and it is represented by ‘d’. Harmonic progression is opposite of arithmetic progression. General term of Geometrical progression is represented as a, ar, ar², ar³………………………… arⁿ. Here, ‘a’ is first term or initial term and 'r' is common ratio.

The nth term of geometric progression is represented as arn-1. And for finding nth term of a series in G.P. we need to be aware about its first term. Recursive relation can also be followed by this kind of G.P. series given as an is equals to (r * a^{n - 1}). Common ratio is not specified as to be positive or negative; it can be anything positive as well as negative. Sum of 'n' terms in a geometric progression series is represented as follows:
S_n = a (1 – rⁿ⁺¹) / (1 - r) when series starts from initial value and for any middle element it is represented as:
S_n = a (r^m – rⁿ⁺¹) / (1 - r) where 'm' is the degree of initial term when it is not zero.
For infinite series in geometric progressions, sum of series is represented as
S_n = a / (1 - r).
A succession of Numbers which are arranged in a definite order according to a certain given rule is called as sequence. A sequence is a finite sequence or infinite according to number of terms in it is finite or infinite respectively. Sequences following certain pattern are called progressions.

By Arithmetic Progression, we Mean that it is a sequence in which each term except first term differs from its preceding term by a constant.

Constant difference is called as common difference of Arithmetic Progressions. To Define Arithmetic Progression, we say that it is the series of numbers which has first term as “a”, common difference “ d” and we define n th term by an. Arithmetic Progression Definition also tells us that n th term of an A.P. with first term 'a' and common difference 'd' is given as follows:
an = a + (n – 1) * d,
Arithmetic Progression Problems solving is always convenient to make different choices based on following :
i) 3 numbers in AP as (a – d), a, (a + d),
ii) 4 numbers in AP as (a – 3d), (a – d), (a + d), (a + 3d),
iii) 5 numbers in AP as (a – 2d), (a – d), a , (a + d), (a + 2d),
iv) 6 numbers in AP as ((a – 5d), (a – 3d), (a – d), (a + d), (a + 3d), (a + 5d).


Arithmetic Progression Proof of two theorems are as follows:

Theorem 1:
Show that the n th term of an AP with first term 'a' and common difference 'd' is given by:
an = a + (n – 1)* d
Proof : Let us consider an AP with first term 'a' and common difference 'd'. Then, AP is given by:
a, a + d, a + 2d, a + 3d, …….
In this AP we have:
First term, a₁ = a = a + 0 * d = a + (1 – 1) * d,
Second term, a₂ = (a + d) = a + (2 - 1) * d,
Third term, a₃ = (a + 2d) = a + (3 – 1) * d,
Thus we say that,
Nth term a_n = a + (n – 1) d,
So we say that a_n = a + (n – 1) d.

Theorem 2:
To show that n th term from end of an AP, with first term 'a', common difference 'd' and last term 'l' is given by [l – (n - 1) d],
Proof: If 'a' is first term, 'd' is common difference and 'l' is last term of given AP
Then the AP is given by: a, a + d, a + 2d, . . . . . , (l – 2d), (l – d), l.
So we say that last term will be = l = l – (1 - 1) d,
Second last term = (l – d) = l – (2 - 1) d,
Third last term = (l – 2d) = l – (3 – 1),
So nth last term will be = l – (n – 1)d.
Progression means increment of quantity in a particular pattern. Progression is of three types:

Arithmetic progression
Geometric progression
Harmonic progression
Here we will focus on Arithmetic Progression.
A sequence of number in which difference between two successive Numbers is a constant is known as arithmetic progression. Representation of arithmetic progression is given as i, i + d, i + 2d, i + 3d, i + 4d …...... i + (n – 1) d, here the value of 'i' represents the initial value and 'd' represents the common difference. For example: 1, 4, 6, 8, 10, 12 here initial value of A.P is 1 and difference between two numbers is 2. Formula for arithmetic progression is given below.
When we want n th term of A.P then formula to find n th term is given by = i + (n – 1) d.
When we want sum of 'n' term in A.P then formula to find n th term is given by = n / 2 (2i + (n – 1) d).
We can find sum of Square of numbers in an A.P then formula to find sum of square of numbers is given by: n (n + 1) (2n + 6) / 6.
If p, q, r are given in A.P then value of 'b' is given as: b = (a + c) / 2. This is all about arithmetic progression formulas. It will be more clear with help of an example:
For example: 'n th' term of a sequence is given as 11, initial term is '1' and common difference is given as '2', which term is that?
Solution: As we know that n th term of sequence is given as: a + (n – 1) d.
Here initial value i = 1, common difference = 2 and n th term sn = 11. We get:
⇒ 11 = 1 + (n – 1) 2,
⇒ 11 – 1 = (n – 1) 2,
⇒ 10/2 = n – 1,
⇒ 5 + 1 = n.
So 6th term of sequence is 11.
When a series is written in such a manner that on writing reciprocal of all terms in same order an Arithmetic Progression is obtained then that series is known as harmonic progression. For example 1 / 3, 1 / 5, 1 / 7, 1 / 9 …… this is a harmonic progression. Now basic harmonic progression formula is shown below:
T_n = 1 / a + (n - 1) * d,
Where Tn is general term of harmonic progression. 'a' is named as first term of concordant arithmetic progression 'd' is common difference and 'n' is number of terms.
There are some important points which will help to solve harmonic progression problems–
1. To solve harmonic progression problems, all terms of harmonic progression first should be written in Arithmetic progression then need to apply all the formulae of A.P. This is the first process to solve any problem of harmonic progressions.
2. Likewise, there is no formula to calculate sum of 'n' terms of harmonic progression so again apply above method means formula of sum of 'n' terms in A.P.
Now general formula of harmonic progression is its harmonic Mean. If there are two entities 'a' and 'b' and 'h' is harmonic mean between these entities, then 'h' can be calculated with help of following formula:
h = 2ab / (a + b),
So, in this case a, h, b will be in harmonic progression and 1 / a, 1 / h and 1 / b will be in arithmetic progression.
Formula to calculate 'n' harmonic means between two quantities 'a' and 'b' can be evaluated as follows–
Suppose that h₁, h₂, h₃… h_n are 'n' harmonic means then 1 / a, 1 / h₁, 1 / h₂ … 1 / h_n, 1 / b will be in arithmetic progression and common difference for this A.P. is given by:
d = a – b / a b (n + 1)
So 'n' arithmetic means can be written between 'a' and 'b' as (1 / a) + d, (1 / a) + 2d, (1 / a) + 3d … (1 / a) + n d which can be simplified as (1 + ad) / a, (a + 2ad) / a, (1 + 3ad) / a … (1 + n a d) / a.
So n harmonic means between a and b will be a / (1 + ad), a / (1 + 2ad), a / (1 + 3ad) … a / (1 + n a d).
The relation between H.P., A.P. and G.P between two entities 'a' and 'b' is given as:
G² = A * H,
Where 'A' is Arithmetic Mean, 'G' is geometric mean and 'H' is harmonic mean and one more Point to be noted is that A, G, H always lie in following order.
A > G > H
That means arithmetic mean is greatest in all means and harmonic mean is least one and basic concept which can be noted in whole article is the definition of harmonic progression which says that harmonic progressions are always reciprocal of terms of arithmetic progression.