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

• All odd positive integers : {1,3,5,7,...} $a_1=1, d=2$
• All positive multiples of 23 : {23,46,69,92,...} $a_1=23, d=23$
• 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

• All positive powers of 2 : {1,2,4,8,...} $b_1=1, r=2$
  
• 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