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Sequences & Progressions : Arithmetic,Geometric and Harmonic [#permalink]
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Sequences & Progressions This post is a part of [GMAT MATH BOOK]
The MOST important from the point of view of GRE is Arithmetic Progressions and then Geometric progressions.
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Definition

Sequence : It is an ordered list of objects. It can be finite or infinite. The elements may repeat themselves more than once in the sequence, and their ordering is important unlike a set

Arithmetic Progressions

Definition
It is a special type of sequence in which the difference between successive terms is constant.

General Term
$$a_n = a_{n-1} + d = a_1 + (n-1)d$$
$$a_i$$ is the ith term
$$d$$ is the common difference
$$a_1$$ is the first term

Defining Properties
Each of the following is necessary & sufficient for a sequence to be an AP :
• $$a_i - a_{i-1} =$$ Constant
• If you pick any 3 consecutive terms, the middle one is the mean of the other two
• For all i,j > k >= 1 : $$\frac{a_i - a_k}{i-k} = \frac{a_j-a_k}{j-k}$$

Summation
The sum of an infinite AP can never be finite except if $$a_1=0$$ & $$d=0$$
The general sum of a n term AP with common difference d is given by $$\frac{n}{2}(2a+(n-1)d)$$
The sum formula may be re-written as $$n * Avg(a_1,a_n) = \frac{n}{2} * (FirstTerm+LastTerm)$$

Examples
1. All odd positive integers : {1,3,5,7,...} $$a_1=1, d=2$$
2. All positive multiples of 23 : {23,46,69,92,...} $$a_1=23, d=23$$
3. All negative reals with decimal part 0.1 : {-0.1,-1.1,-2.1,-3.1,...} $$a_1=-0.1, d=-1$$

Geometric Progressions

Definition
It is a special type of sequence in which the ratio of consequetive terms is constant

General Term
$$b_n = b_{n-1} * r = a_1 * r^{n-1}$$
$$b_i$$ is the ith term
$$r$$ is the common ratio
$$b_1$$ is the first term

Defining Properties
Each of the following is necessary & sufficient for a sequence to be an GP :
• $$\frac{b_i}{b_{i-1}} =$$ 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 : $$(\frac{b_i}{b_k})^{j-k} = (\frac{b_j}{b_k})^{i-k}$$

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 $$b_1*\frac{r^n - 1}{r-1}$$
If an infinite GP is summable (|r|<1) then the sum is $$\frac{b_1}{1-r}$$

Examples
1. All positive powers of 2 : {1,2,4,8,...} $$b_1=1, r=2$$
2. All negative powers of 4 : {1/4,1/16,1/64,1/256,...} $$b_1=1/4, r=1/4, sum=\frac{1/4}{(1-1/4)}=(1/3)$$

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 HP

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 :
$$\frac{1}{2} * (\frac{1}{a} + \frac{1}{b}) = \frac{1}{HM(a,b)}$$
Or in other words :
$$HM(a,b) = \frac{2ab}{a+b}$$

Each progression provides us a definition of "mean" :

Arithmetic Mean : $$\frac{a+b}{2}$$ OR $$\frac{a1+..+an}{n}$$
Geometric Mean : $$\sqrt{ab}$$ OR $$(a1 *..* an)^{\frac{1}{n}}$$
Harmonic Mean : $$\frac{2ab}{a+b}$$ OR $$\frac{n}{\frac{1}{a1}+..+\frac{1}{an}}$$

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

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

Example :
Let a=50 and b=2,
then the AM = (50+2)*0.5 = 26 ;
the GM = sqrt(50*2) = 10 ;
the HM = (2*50*2)/(52) = 3.85
AM > GM > HM
AM*HM = 100 = GM^2

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_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

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Re: Sequences & Progressions : Arithmetic,Geometric and Harmonic [#permalink]
Thanks Re: Sequences & Progressions : Arithmetic,Geometric and Harmonic   [#permalink] 31 Aug 2019, 02:09
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