Arithmetic progression

shreyadas

Shreya Das
In mathematics, 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 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference 2.
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence is given by:

and in general

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression.
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
• Positive, the members (terms) will grow towards positive infinity.
• Negative, the members (terms) will grow towards negative infinity.
Contents
• 1 Sum
• 2 Product
• 3References
• 4 See Also
• 5External links


Sum
The sum of the members of a finite arithmetic progression is called an arithmetic series.
Expressing the arithmetic series in two different ways:


Adding both sides of the two equations, all terms involving d cancel:

Dividing both sides by 2 produces a common form of the equation:

An alternate form results from re-inserting the substitution: an = a1 + (n − 1)d:

In 499 CE Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya (section 2.18) .[1]
So, for example, the sum of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up to the 50th term is





Product
The product of the members of a finite arithmetic progression with an initial element a1, common differences d, and n elements in total is determined in a closed expression

where denotes the rising factorial and Γ denotes the Gamma function. (Note however that the formula is not valid when a1 / d is a negative integer or zero.)
This is a generalization from the fact that the product of the progression is given by the factorial n! and that the product

for positive integers m and n is given by

Taking the example from above, the product of the terms of the arithmetic progression given by an = 3 + (n-1)(5) up to the 50th term is







Arithmetic operations
The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, and logarithmic functions. Arithmetic is performed according to an order of operations. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws, is called a field.
Addition (+)
Main article: Addition
Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends or terms, into a single number, the sum of the numbers.
Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting.
Addition is commutative and associative so the order the terms are added in does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number yields that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself yields the additive identity, 0. For example, the opposite of 7 is −7, so 7 + (−7) = 0.
Addition can be given geometrically as in the following example:
If we have two sticks of lengths 2 and 5, then if we place the sticks one after the other, the length of the stick thus formed is 2 + 5 = 7.
Subtraction (−)
Main article: Subtraction
Subtraction is the opposite of addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend. If the minuend is larger than the subtrahend, the difference is positive; if the minuend is smaller than the subtrahend, the difference is negative; if they are equal, the difference is zero.
Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is a − b = a + (−b). When written as a sum, all the properties of addition hold.
There are several methods for calculating results, some of which are particularly advantageous to machine calculation. For example, digital computers employ the method of two's complement. Of great importance is the counting up method by which change is made. Suppose an amount P is given to pay the required amount Q, with P greater than Q. Rather than performing the subtraction P − Q and counting out that amount in change, money is counted out starting at Q and continuing until reaching P. Although the amount counted out must equal the result of the subtraction P − Q, the subtraction was never really done and the value of P − Q might still be unknown to the change-maker.
See also: Method of complements





Multiplication (× or • or *)
Main article: Multiplication
Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, sometimes both simply called factors.
Multiplication is best viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number, say x, greater than 1 is the same as stretching everything away from zero uniformly, in such a way that the number 1 itself is stretched to where x was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards zero. (Again, in such a way that 1 goes to the multiplicand.)
Multiplication is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 yields that same number. Also, the multiplicative inverse is the reciprocal of any number (except zero; zero is the only number without a multiplicative inverse), that is, multiplying the reciprocal of any number by the number itself yields the multiplicative identity.
The product of a and b is written as a × b or a • b. When a or b are expressions not written simply with digits, it is also written by simple juxtaposition: ab. In computer programming languages and software packages in which one can only use characters normally found on a keyboard, it is often written with an asterisk: a * b.



Division (÷ or /)
Main article: Division (mathematics)
Division is essentially the opposite of multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient is greater than one, otherwise it is less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.
Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1/b. When written as a product, it obeys all the properties of multiplication
















Derivative Rules
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Introduction

In economics and commerce, the concept of rate is very commonly used and is important. For example: we are interested in finding out how fast the price of a commodity or a share of changing with time. We are interested in knowing how fast (or slow) the supply quantity of a product in the market changes with change in its price. We are interested in finding out the rate at which the total revenue changes with respect to change in quantity sold

In general , if y = f(x) is a function involving a dependent variable y which depends on an independent variable x , then we may be interested in finding out the rate of change of y values with respect to a change in x values .This rate is called the derivative . It gives the instantaneous rate of the curve y = f(x)
At the instant or at the point x . It is denoted by dy/dx or by f’(x)

There are some rules of derivatives which help in finding derivatives of some more complicated functions. If one can find the derivative of a function at each point x , we call it differential function . if a function is a sum , difference , scalar multiple , product or a quotient of differentiable functions , then we can use the rules of differentiation


Rules of derivative


The Constant Rule


The derivative of a constant function is 0. That is, if c is a real number, then d/dx[c] = 0.



The Sum and Difference Rules
The sum(or difference) of two differentiable functions is differentiable and is the sum(or difference) of their derivatives.
d/dx[f(x) + g(x)] = f'(x) + g'(x)

d/dx[f(x) - g(x)] = f'(x) - g'(x)




The Constant Multiple Rule
If f is a differentiable function and c is a real number, then cf is also differentiable and d/dx[cf(x)] = cf'(x)




The Power Rule
If n is a rational number, then the function f(x) = xn is differentiable and d/dx[xn] = nxn-1



The Product Rule
The product of two differentiable functions, f and g, is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first.
d/dx[f(x)g(x)] = f(x)g'(x) + g(x)f'(x)




The Quotient Rule
The quotient f/g, of two differentiable functions, f and g, is itself differentiable at all values of x for which g(x) does not = 0. Moreover, the derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by the square of the denominator.
d/dx[ f(x)/g(x) ] = (g(x)f'(x) - f(x)g'(x)) / [g(x)]2 g(x) does not = 0


The Chain Rule
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and d/dx[f(g(x))] =f'(g(x))g'(x)

The General Power Rule
If y = [u(x)]n, where u is a differentiable function of x and n is a rational number, then d/dx = [un] = nun-1u'.
 

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