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# Progression

=When the terms of a sequence and Series are written under some speciﬁc conditions, then it would be known as progression.

There are Three Types of Progression:

Arithmetic Progression (AP)= It is a special type of sequence in which the difference between successive terms is constant.

an= an-1 +d = a1+(n-1)

ai is the ith term

a1 is the first term

Defining Properties

Each of the following is necessary & sufficient for a sequence to be an AP :

• a2-a2-1 = Constant
• If you pick any 3 consecutive terms, the middle one is the mean of the other two
• The general sum of a n term AP with common difference d is given by = n/2 ( 2a+(n-1)d)
• Note-: When we have to assume 3 terms , then (a-d), a,  and (a+d)
• When we have to assume 5 terms, then (a-2d), (a-d), a, (a+d), and (a+2d)

Geometric Progression (GP)

Definition

It is a special type of sequence in which the ratio of consecutive terms is constant.

General Term

nth term of a geometric progression

•  First term t1 = a = ar1-1
•  Second term t2 = ar = ar2-1
• Third term t3 = ar2 = ar3-1
• Fourth term t4 = ar3 = ar4-1
• nth term  tn = arn-1

bn = bn-1*r= a1*rn-1

bi is the ithe term

r is the common ratio

b1 is the first term

Defining Properties Each of the following is necessary & sufficient for a sequence to be an AP

• bi/bi-1 = Constant
• If you pick any 3 consecutive terms, the middle one is the geometric mean of the other two
• For all i,j > k >= 1 :

Harmonic Progression (HP)

Definition It is a special type of sequence in which if you take the inverse of every term, this new sequence forms an AP

Important Properties Of any three consecutive terms of a HP, the middle one is always the harmonic mean of the other two, where the harmonic mean (HM) is defined as :

## MEAN

Each progression provides us a definition of “mean”

• Arithmetic Progression= a+b/2
• Geometric Progression= √ab
• Harmonic Progression= 2ab/a+b

For all non-negative real numbers : AM >= GM >= HM

In particular for 2 numbers : AM * HM = GM * GM

# Misc. Notes

A subsequence (any set of consecutive terms) of an AP is an AP

A subsequence (any set of consecutive terms) of a GP is a GP

A subsequence (any set of consecutive terms) of a HP is a HP

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