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Investigating Sequences and Series

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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

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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 between two numbers is 4. Now we will discuss how to find the sum of 'n' terms in series arithmetic.
Using formula we can calculate arithmetic series. Formula to calculate sum of 'n' terms in Arithmetic Progression is:
sum of n terms = n / 2 (2i + (n – 1) d).

Let’s understand this formula with help of an example:
Suppose we have to calculate sum of first 20 terms of arithmetic progression: 8, 3, -2..........and so on,
As we know that the sum of 'n' terms is given as:

sum of n terms = n / 2 (2i + (n – 1) d).
Here 'i' shows the initial value and value of a = 8, 'd' is common difference, value of d = 3 – 8 = -5, and value of 'n' = 20. So put these value in formula, we get:
sum of n terms = n / 2 (2i + (n – 1) d).
sum of n terms = 20 / 2 (2 * 8 + (20 – 1) (-5)),
sum of n terms = 10 (16 + (19) (-5)),
sum of n terms = 10 [16 – 95],
sum of n terms = 10 [-79],
sum of n terms = - 790,
So sum of first 20 terms is - 790. This is all about Arithmetic series.
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 common Ratio is given as "f", series can be written as:

n, n * f, n * f2, n * f3, n * f4 and so on.

For example, the series 2, 16, 128, 1024 is a Geometric Series with the first term as 2 and common ratio is given as 16 / 2 or 1024 / 128 = 8. So, we just need to be available with 1st term and common factor to generate the complete series.
Questions of Geometric Progression are solved by following these steps: First note down the given facts about the succession. The format can be as follows: The 1st term and common proportion or the 1st term and the following sequential term of the sequence. For instance, say if the first term is given as 12 and next term as 24 or it has been given that 's' is any term of the succession and its Position is 'i' in the series with ratio 'f'.
Formula for first term of the geometric sequence:

n = L / [f (m – 1)],

Where, 'L' represents the arbitrary term of the sequence with its position in the arrangement as 'm' and common ratio 'f'. So, if 10th term of the sequence is 10240 i.e. m = 10 and L = 1024. What would be the first term of such GP with f = 2?
n = 1024 / 29 = 2.


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Fractals

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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. This can also be defined as spreading pattern which repeats itself on each of the scale. Mathematically, fractal can be defined as a Set of nodes or points whose dimensions are spread over dimension of its topology. An infinite fractal graph can be obtained by repeating it for infinite time through space.

Fractal is an n-dimensional pattern. Fractals are those patterns that cannot be limited to a geometric pattern. When we take a line and process it through n-dimensional space in such a way that it generates another pattern and this pattern is called fractal. Pattern of fractals can be used in various fields such as images or structures or sounds. Naturally, there may be lot of fractal patterns. Fractal patterns are designed with various degrees.

Another term multi fractals is also used, to obtain multi fractals, first test the source's distribution over area then process and compute the various dimensions of the parts giving the similar amount and finally graph the oriented dimensions. Graph is plotted considering the dimensions as its variation with amount. Another term in case of fractal Geometry is used which is called as fractal trees. When various branches are combined, they are called as trees. Similarly, fractal Tree is defined when several fractals are combined together. Under the study of fractal trees, a small description of dimensions of various views of several parts is done.

Infinite Geometric Series

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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 + ….. + i1rn – 1. (If -1 < r < 1). Formula to find the sum of infinite geometric series is given as:

=> Sn = i1 / 1 – r (| r | < 1).

Now we will see how to calculate the sum of an infinite geometric series with help of an example.
Suppose we have a series 5 + 2.5 + 1.25 + 0.625 + 0.3125.....
In this first term is given as i1 = 5 (initial value) and common ratio is r = 0.5

So value of common ratio lies between -1 and 1, series will converge to some value. Let's discuss the sum of first few terms.
=> i1 = 5,
=> i1 + i1r = 5 + 2.5 = 7.5,
=> i1 + i1r + i1r2 = 5 + 2.5 + 1.25 = 8.75,
=> i1 + i1r + i1r2 + i1r3 = 5 + 2.5 + 1.25 + 0.625 = 9.375,
So the sum of 5 terms = 9.84375,
Sum of 6 terms = 9.921875,
Sum of 7 terms = 9.9609375,
Sum of 8 terms = 9.98046875,
Sum of 9 terms = 9.990234375,
Sum of 10 terms = 9.955117188,
Sum of 11 terms = 9.997558594,
Sum of 12 terms = 9.998779297,

Sum of 6 terms = 9.999389648, continue this process till last, here we will discuss that sum is not more than value of 10. Using formula we will get same result. As we discussed above that formula is given as:
=> Sn = i1 / 1 – r,
=> Sn = 5 / 1 – 0.5,
On further solving we get:
=> Sn = 5 / 0.5 = 10,
So value of Sn we get is 10.
In this way we can calculate sum of geometric infinite series.

Geometric Series

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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 very important role in Calculus. It is also applied in convergence of series. It has applications in many fields such as physics, engineering, computer science, finance etc.
Now we will see common ratio used in geometrical series.
According to definition of geometric series table shown below has different common ratios.

Common Ratio

Example

10

3 + 30 + 300 + 4000 + 40, 000 + …..

1 / 3

9 + 3 + 1 + 1 / 3 + 1 / 9 + 1 / 81 + ……

1 / 10

7 + 0.7 + 0.07 + 0.007 + 0.0007 + …..

1

4 + 4 + 4 + 4 + 4 + 4 + …..

-1 / 2

1 – 1 / 2 + 1 / 4 – 1 / 8 + 1 / 16 – 1 / 32 + ..

-1

5 – 5 + 5 – 5 + 5 - ……..

-2

3 – 6 + 12 – 24 + 48 – 96 + …..

In this way we can find the common ratio in geometric series. Behavior of terms totally depends on common ratio 'u'.
If value of ‘u’ is between -1 to +1. In above given case value of ‘u’ is one half and sum of series is equals to one. This is all about geometrical series.

Binomial Theorems

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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 is quite difficult, so to solve these types of expressions we use binomial theorems. This theorem states that it is possible to expand (x + y)n expression, for any value of 'n'.

Formula of binomial theorem:-
(x + y)n = sigma k = 0 → n (n choose k) x(n - k) yk ---- eq1,
Here n and k are called binomial coefficients, we will solve this equation until power of 'x' reaches zero and power of 'y' reaches 'n' .
'n' choose 'k' means (nCk) we have a formula to solve this →
n choose k = nCk = n! / (n - k)! k!,
We know that n! = multiplication of all Numbers from 1 to n.

Eg:- 5 choose 2 = 5! / (5 - 2)! ( 2)! => 5*4*3*2*1 / (3*2*1) (2*1) => 10.
So put the value of nCk in the eq1, equation 1 becomes:-
(x + y)n = sigma k= n ( n! / ( n- k)! k!) ( x (n-k) y k).
If we want to expand this expression (x + y)6 using binomial theorem, then we get,
(x + y)6 = x6 + 6 x5 y + 15 x4 y2 + 20 x3 y3 + 15x2 y4 + 6x y5 + y6.