Series and Progressions

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Arithmetic Progression: 

An Arithmetic Progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. 

For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2.

Its general form can be given as a, a+d, a+2d, a+3d,...

If the initial term of an arithmetic progression is a and the common difference of successive members is d, then the 
nth term of the sequence (a_n) is given by: 
an = a + (n - 1)d

and in general 

Nth Term of A.P. is An = am + (n - m)d

The sum of the members of a finite arithmetic progression is called an arithmetic series and given by, 

Sum of N terms of an A.P. is Sn/2 [2a + (n - 1)d]  = n/(a + l)

Arithmetic mean: 

When three quantities are in AP, the middle one is said to be the Arithmetic Mean (AM) of the other two, thus a is the AM of (a-d) and (a+d).
Arithmetic mean between two numbers a and b is given by, 
AM =  (a+b)/2

Geometric progression:

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. 

The general form of a geometric sequence is a, ar, ar2,ar3,ar4,…

geometric series is the sum of the numbers in a geometric progression.

Let a be the first term and r be the common ratio, an nth term, n the number of terms, and Sn be the sum up to n terms:

The n-th term  is given by, 
an = arn-1

The Sum up to n-th term of Geometric progression (G.P.)  is given by,

If r > 1, then

S= a(rn-1)/(r-1)

if r < 1, then
Sn = a(1-rn)/(1-r)

Sum of infinite geometric progression when r<1: 

Sn = a/(1-r)

Geometric Mean (GM) between two numbers a and b is given by, 

GM = sqrt ab 

Some useful results on number series:

Sum of first n natural numbers is given by

S = 1 + 2 + 3 + 4 +....+n 
S = n/2 * (n+1) 

Sum of squares of the first n natural numbers is given by

S = 12 + 22 + 32 +....+n2
S = [{n(n+1)(2n+1)}/6 ]

Sum of cubes of the first n natural numbers is given by

S = 13 + 23 + 33 +....+n3
S = [{n(n+1)}/2]

Sum of first n odd natural numbers 

S = 1 + 3 + 5 +...+ (2n-1)
S = n2

Sum of first n even natural numbers S = 2 + 4 + 6 +...+ 2n

S = n(n+1)

1) If we are counting from n1 to n2 including both the end points, we get (n2-n1) + 1 numbers.
e.g. between 12 and 22, there is (22-12) +1 = 11 numbers (Including both the ends).

2) In the first n, natural numbers:
i) If n is even
There are n/2 odd and n/2 even numbers 
e.g from 1 to 40 there are 25 odd numbers and 25 even numbers.
ii) If n is odd
There are (n+1)/2 odd numbers, and (n-1)/2 even numbers
e.g. from 1 to 41, there are (41+1)/2= 21 odd numbers and (41-1)/2 = 20 even numbers.

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Series and Progressions Series and Progressions Reviewed by Admin on 3:09:00 PM Rating: 5
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