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 :
\(HM(a,b) = \frac{2ab}{a+b}\)

APs, GPs, HPs : Linkage

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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