# 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

*n*th term of the sequence () is given by:

*a*_{n}= a + (n - 1)d

and in general

###
*Nth Term of A.P. is A*_{n} = a_{m} + (n - m)d

*Nth Term of A.P. is A*

_{n}= a_{m}+ (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 S*_{n }= ^{n}/_{2} [2a + (n - 1)d] = ^{n}/_{2 }(a + l)

*Sum of N terms of an A.P. is S*

_{n }=^{n}/_{2}[2a + (n - 1)d] =^{n}/_{2 }(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}_{}

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_{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, ar*^{2},ar^{3},ar^{4},…

A

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

*a*be the first term and*r*be the common ratio,**a**nth term, n the number of terms, and_{n}**S**be the sum up to n terms:_{n}###
The **n****-th term** is given by,

*a*_{n} = ar^{n-1}

**n**

*a*

_{n}= ar^{n-1}^{}

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

**n**

*If r > 1, then*

*S*_{n }= a(r^{n}-1)/(r-1)

*if r < 1, then*

*S*_{n}= a(1-r^{n})/(1-r)

Sum of infinite geometric progression when r<1:

*S*_{n}= a/(1-r)

###
**Geometric M****ean** (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 = 1

^{2}+ 2^{2}+ 3^{2}+....+n^{2}

^{S = [{n(n+1)(2n+1)}/6 ]}

^{}#### Sum of cubes of the first n natural numbers is given by

S = 1

^{3}+ 2^{3}+ 3^{3}+....+n^{3}

^{S = [{n(n+1)}/2]}

^{}#### Sum of first n odd natural numbers

S = 1 + 3 + 5 +...+ (2n-1)

*S = n*^{2}

^{}#### Sum of first n even natural numbers S = 2 + 4 + 6 +...+ 2n

*S = n(n+1)*

**Note:**
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
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