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# Investigating Sequences And Series

Three types of sequences are there i.e. A.P. (Arithmetic Progression), G.P. (Geometric Progression) and Harmonic Progression (H.P.), here we will be investigating sequences of these three types.
Progression is defined as sequence of terms that increases in a particular pattern.
1.               Arithmetic sequence
2.               Geometric sequence
3.               Harmonic sequence.

Let's have small introduction about these sequences.
Sequence of Numbers in which there is a constant difference between any two consecutive terms is known as Arithmetic Progression. A.P. Can be given as a, a + d, a + 2d, a + 3d, a + 4d …...... a + (n – 1) d, here value of 'a' represents the initial value and 'd' represents common difference. For example: 6, 7, 8, 9, 10 here initial value of A.P is 6 and the common difference between two number is also 1. Formula for arithmetic progression is given as:

1.               If we calculate n th term of A.P. then we use formula that is given as = a + (n – 1) d.
2.               If we calculate sum of 'n' terms in A.P. then we use formula which is given as = n / 2 (2a + (n – 1) d).
Geometric progression: - Geometric progression is a sequence of numbers in which Ratio of two successive numbers is constant. Geometric sequence ’s’ can be given as: a, ar, ar2, ar3......., ar (n – 1). Here initial number is denoted as 'a' and common ratio is denoted as 'r'. For example: 1, 2, 4, 8, 16, 32. Here initial number is 1 and common ratio is 2.
Harmonic sequences: - If in the sequence difference of successive denominators of Fractions is same then it is called as harmonic sequence. Harmonic sequences are given as: a, a/d, a/(a + d), a/(a + 2d),..... a/a + nd).

## Arithmetic Series

Sequence of Numbers in which difference between any two consecutive numbers is constant is known as arithmetic series. Arithmetic progression is denoted as i, i + d, i + 2d, i + 3d, i + 4d …...... i + (n – 1) d, here 'i' denotes initial value and 'd' denotes the common difference. For example: 20, 24, 28, 32, 36, 40 here initial value of A.P is 20 and difference betwee...Read More

## Geometric Sequence

Geometric Sequences are a form of mathematical successions in which every single number in the succession is acquired by multiplying the previous number by a persistent factor. Other way it can be stated that in a geometric sequence there is a permanent fraction sustained among the two serial terms in the sequence. Suppose we have a GP in which first value is "n" and ...Read More

## Fractals

Fractal can be defined as distributed geometric shape which can be further broken into several parts. Each of the part is a copy of whole fractal. This is named so since its dimensions are fractal. Fractals are self identical Patterns that are similar whether they are far or close. Fractals are similar at every scale that is all parts of fractals are identical to each other. Th...Read More

## Infinite Geometric Series

Geometric sequence is a process of arranging the Numbers in such a way that Ratio of two successive numbers is constant. In case of geometric sequences initial number is given as 'i' and common ratio is given as 'r'. Now we will see process of calculating sum of infinite Geometric Series. Infinite Geometric Progression is given as: i1 + i1r + i1r2 + i1r3 + ….. ...Read More

## Geometric Series

Geometric series is also called as Geometric Progression, in Geometric Sequence Ratio of consecutive terms in series is constant and it is known as common ratio. For example: 1 / 3 + 1 / 6 + 1 / 12 + 1 / 24 + ……and so on. It is an example of geometrical series, it is so because every consecutive term can be found by multiplying the previous term by 1 / 2. It plays a ver...Read More

## Binomial Theorems

Binomial theorem is the best way to expand any expression which is written in the form of binomial expression. If we want to multiply any expression for two or three times then we can multiply them but multiplication of power for example 20 is very difficult. For example: If we want to solve (2x - 1)2 then we can easily do this, but solving this expression (2x - 1) 10 ...Read More