Sequences & ProgressionsThis 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.--------------------------------------------------------

DefinitionSequence : 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 ProgressionsDefinitionIt 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 PropertiesEach 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 : \(\frac{a_i - a_k}{i-k} = \frac{a_j-a_k}{j-k}\)

SummationThe 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 ProgressionsDefinitionIt 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 PropertiesEach 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 : \((\frac{b_i}{b_k})^{j-k} = (\frac{b_j}{b_k})^{i-k}\)

SummationThe 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 ProgressionsDefinitionIt is a special type of sequence in which if you take the inverse of every term, this new sequence forms an HP

Important PropertiesOf 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 : LinkageEach 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

--------------------------------------------------------

_________________

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