Method of differences

There are many occurrences when we have certain numbers and we have to find relation between them. The relation is usually in form of expressions. The expression can contain log function, exponential function and algebraic functions.The simplest among them is to find polynomial expressions.In this post I will describe how to find functions when we know a series which has its terms separated by certain constant. The method is similar to differentiation but a lot different from them.

Suppose we have the numbers (from a function) like given in the following table

1	6	15	28	45	66	91
We find the difference of successive term
5		9	13	17	21	25
We again find the difference of successive term
4			4	4	4	4
We find the differences till we get a constant term for all

As we got the constant term so we can create a table of possible polynomials for all. We start from the bottom.

c''	4			4	4	4	4
a'x+b'	5		9	13	17	21	25
ax2+bx+c	1	6	15	28	45	66	91

We got a quadratic because we had three rows in the table.
Now we have
f(x) =ax²+bx+c
f(0) = c = 1
f(1) = a + b + c = 6
f(2) = 4a + 2b + c = 15
from f(0): c =1
from f(1): a + b = 5
from f(2): 4a + 2b = 14
so solving the simultaneous equations a = 2 and b = 3 hence the function is f(x) = 2x² + 3x + 1.
Above it must be known from where the values start for x and what is their differences in each step.

Mathematics: Need and areas

"Mathematics quantify our life"
The essential of mathematics is clearly visible from the following paragraph.

A person wakes up. He wants to know what is the elapsed time from the sunrise. He makes a sundial and looks at the relative position. After sometime he starts his journey towards his work place. He reaches there and finds that his one customer waited for him and left for somewhere else. He thought he must have known the time accurately.

Accuracy is the cause of all mathematics. A man can find the amount of bulk investment but how to distribute it in chunks is taught by mathematics. One can find the shortest path from one place to other but what time it will take to reach that place is told by mathematics. One can pass a vehicle moving ahead of it. But how much time it will take to pass it is told by mathematics.There are many such cases in which the estimation is replaced by accuracy with the help of mathematics.

Mathematics is used in all spheres of life. It is used by a boy to the adult and at last by a old person. A shopkeeper uses it keep record of his profits and losses. A carpenter uses it to measure the extent of his furniture. A businessman uses it to increase his profits. A sportsperson uses it for accuracy. A scientist uses it in different fields of research. Such is the extent of mathematics that if you need accuracy then there is no substitute for it.

There are many branches of mathematics such as calculus, algebra, analysis, geometry, trigonometry, probability and statistics, etc.