Series and Progressions

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:

a_n = a + (n - 1)d

and in general

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)

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,…

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_n nth term, n the number of terms, and S_n be the sum up to n terms:

The n-th term is given by,
a_n = ar^n-1

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

If r > 1, then

S_n= a(rⁿ-1)/(r-1)

if r < 1, then

S_n = a(1-rⁿ)/(1-r)

Sum of infinite geometric progression when r<1:

S_n = 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)

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