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is the first term
Defining Properties
Each of the following is necessary & sufficient for a sequence to be an AP :
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 :
Summation
The sum of an infinite GP will be finite if absolute value of r < 1
The general sum of a n term GP with common ratio r is given by
If an infinite GP is summable (|r|<1) then the sum is
Examples
1. All positive powers of 2 : {1,2,4,8,...}
2. All positive odd and negative even numbers : {1,‐2,3,‐4,...}
3. All negative powers of 4 :
{1/4,1/16,1/64,1/256,...}
Harmonic Progressions
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 :
Or in other words :
APs, GPs, HPs : Linkage
Each progression provides us a definition of "mean" :
Arithmetic Mean :
Geometric Mean :
OR
OR
Harmonic Mean :
OR
For all non‐negative real numbers : AM >= GM >= HM
In particular for 2 numbers : AM * HM = GM * GM
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