Sum of Positive Integers

Introduction about sum of positive integers:

The integers are formed by the natural numbers including 0 (0,1,2,3…..) together with the negatives  of the non-zero natural numbers (-1, -2, -3, ...) The positive integers are 0,1,2,3…..

To find the sum of consecutive positive integers we have to use the following formula,

                 `sum_(i=1)^n`n = `(n (n+1))/(2)`  where n is the positive integer

Properties of integer

Addition (or) sum: If we have added two positive integers the result will be a positive integer. If negative integer involved in addition means the result is based on the biggest number.

Commutative property: a+b= b+a
Associative property:a+(b+c)=(a+b)+c
Identity property: a+1= 1+a

Algebra is widely used in day to day activities watch out for my forthcoming posts on how to solve quadratic inequalities and algebra linear equations. I am sure they will be helpful.

Lets Solve some Examples on Sum of Positive Integers

Pro 1 :Find the sum of consecutive positive integers from 1 to 25

Solution:From the formula  `sum_(i=1)^n`n  = `(n (n+1))/(2)`

Here n =25  `sum_(i=1)^25`25  = `(25 *(25+1))/(2)`

                                       = `(25 * 26 )/(2)`

                                       =`(650)/(2)`

                                       = 325

The answer for sum of consecutive integer from 1 to 25 is 325

Pro 2:Find the sum of consecutive positive integers from 1 to 50

Solution:From the formula  `sum_(i=1)^n`n  = `(n (n+1))/(2)`

Here n =50  `sum_(i=1)^50`50  = `(50 *(50+1))/(2)`

                                       = `(50 * 51)/(2)`

                                       =`(2550)/(2)`

                                       = 1275

The answer for sum of consecutive integer from 1 to 25 is 1275

Pro 3:Find the sum of positive integer 5 + 10.

The symbol plus +, represents the positive direction.
So, To find 5+10, number line starts from 5,  go to the  positive direction and move 10 units frontward we will get the 15

5+10=15

The answer is 15

Some more Problems on Sum of Positive Integers

Pro 4: Find the sum of consecutive positive integers from 1 to100

Solution:

From the formula  `sum_(i=1)^n`n  = `(n (n+1))/(2)`

Here n =100 =  `sum_(i=1)^100`100 = `(100 *(100+1))/(2)`

                                       = `(100 * 101)/(2)`

                                       =`(10100)/(2)`

                                       = 5050

The answer for sum of consecutive integer from 1 to 100 is 5050

Pro 5:Find the sum of consecutive positive integers from 1 to 45

Solution:From the formula `sum_(i=1)^n`n  = `(n (n+1))/(2)`

Here n =45  `sum_(i=1)^45`45  = `(45 *(45+1))/(2)`

                                       =`(45 * 46)/(2)`

                                       =`(2070)/(2)`

                                       = 1035

The answer for sum of consecutive integer from 1 to 45 is 1035

Introduction of add polynomials

Introduction of add polynomials:

          A polynomial is an expression formed by variables and constant, using the  fundamental operation addition, subtraction, multiplication, and non-negative, number exponents. For example, x 2 - 4x + 5 is a polynomial. An algebraic expression, ax 2 is called a monomial in x, x is a variable a. The number a is called the coefficient of x^2 , the degree of the monomial. The sum of two monomials is called a binomial. The sum of a limited number of monomials in x is called a polynomial in x.

Please express your views of this topic solve a math problem by commenting on blog.

Polynomials and Methods for Adding

The monomials in a polynomial are called the terms and the highest degree of the terms, the degree of the polynomial. The coefficient of the highest power of x in a polynomial is called the leading coefficient of the polynomial. For example, 3x – 1 + 5x^2 + 6x^5 – 4x3 is a polynomial of degree 5 with terms 2x, –1, 5x^2, 6x^5 and –4x3 and most important coefficient 6. The terms of a polynomial are real numbers, they are rearranged using the property of real numbers

A first degree polynomial is called a linear polynomial. A second degree polynomial is called a quadratic polynomial. A degree polynomial is third called as cubic polynomial and a degree polynomial is four called as bi-quadratic polynomial. Since the terms of a polynomial are numbers, all polynomials represent numbers. So we can add, subtract, multiply and divide polynomials. The following method used to add the polynomials.

Adding Polynomials Examples

Ex 1: Find the sum or add of 3x^4 – 4x^2 + 5x + 3 and 4x + 6x3 – 6x^2 – 2 polynomials.

Sol :  Using the properties of real numbers, we add the polynomial

(3x^4 – 4x^2 + 5x + 3) + (6x3 – 6x^2 + 4x – 2) = 3x^4 + 6x3 – 4x^2 – 6x^2 + 5x + 4x + 3 – 2

= 3x^4 + 6x3 – (4+6)x^2 + (5+4)x + 1

= 3x^4 + 6x3 – 10x^2 + 9x + 1.

The following method is useful in add two polynomials

3x^4 + 0x3 – 4x^2 + 5x + 3

0x^4 + 6x3 – 6x^2 + 4x – 2

_____________________

3x^4 + 6x3– 10x^2 + 9x + 1

_____________________

Graphing Linear Equations and Functions

Introduction

Linear equations are relation between two variables. Generally in form of ax+by+c = 0 or ax+by = c. We can see linear functions also. These are nothing but similar to linear equations. Generally a function can be in form of f(x) = ax + b, we can write this function as y = ax + b also.

Now graphing linear equations and functions are very simple. Since these are a straight lines, you have to just find two points through which these lines passes and once you find the point you can draw a line meeting both the point. Here are steps for graphing equations and functions.

Steps for Graphing Linear Equations and Functions

Step1: Put x=0, you will get value of y, lets say y = b, first point will be (0,b)

Step2: put y=0, you will get value of x, lets say x = a, second point will be (a,0)

Step3: Plot both the points and draw a line meeting both the points and extend this line beyond these points.

In this way you will get required graph of linear equation and functions.

Following examples help you in understanding these steps and similar problems.

Examples for Graphing Linear Equations and Functions

Ex:1 Graph the linear equation: 2x + y = 6

Put x=0, we get y =6

first point wil be (0,6)

Put y=0, we get x=3

second point will be (3,0)

Now plot these points and draw the required graph of this linear equation.

Understanding algebra 2 help online is always challenging for me but thanks to all math help websites to help me out.

Ex:2 Graph the function f(x) = 4x + 4

put in the form of y = 4x + 4

Put x = 0, we get y = 4, First point will be (0,4)

Put y = 0, we get x=-1, second point will be (-1,0)

Now plot the points and draw the graph.

Introduction of sample proportion

Introduction of sample proportion:

          Two equal ratios form a proportion. Thus, if a:b=c:d then a,b,c,d are said to be sample proportional. We write this as, a:b::c:d. This is read as “a is to be is as c is to d”A and b are called the extremes and b an c are called the means. The terms d is called the fourth sample proportional to a,b,c.

Types of Sample Proportions:

Continued proportion: quantities a,b,c,d,…. Are said to be in continued proportion if, a/b=b/c=c/d….. i.e., a:b=b:c=c:d=……………….

Mean proportional:  consider a,b, and c are in continued proportion, Proportion will be represented as a:b::b:c. Here, b is called the mean proportional of a and c..

Also c is called the third proportional to a and b.. If a,b, c are in continued proportion, then ac:b² . Hence, if three quantities are in continued proportion, then the product of extremes is equal to the square of the mean.

Note: The product of the extremes is equal to the product of the means.

Inadvertent: If a:b::c:d, then, b:a::d:c. for, a/b=c/d gives a/c=b/d. This operation is called inadvertent. It says that if quantities are proportion they are also proportional when we take it inversely.

Alternendo: if a:b::c:d then a:c::b:c.

For, a/b=c/d gives a/c=b/d. This operation is called alternendo. It says that if four quantities are proportional, they are also proportional when taken alternatively.

Componendo: If a:b::c:d, then (a+b):b:(c+c):d.

Given that, a/b=c/d,

So,(a/b)+1=(c/d)+1

Or, (a+b)/b=(c+d)/d

This operation is called componendo.

Dividendo: If a:b::c:d,

Then (a-b):b::(c-d):d.

Given that, a/b=c/d,

So,(a/b)-1=(c/d)-1

Or, (a-b)/b=(c-d)/d.

This operation is called dividendo.

Componendo and Dividendo:

If a:b::c:d,

Then (a+b):(a-b)::(c+d):(c-d).

Using componendo and dividendo respectively, we get

(a+b)/b=(c+d)/d   -------------------------1

(a-b)/b=(c-d)/d.    -------------------------2

Dividing the respective sides of 1 by those of 2 we get

(a+b)/(a-b)=(c+d)/(c-d)

This operation is called componendo and dividendo.

Examples for Finding the Sample Proportion:

Ex 1  : 2/5=4/10

Sol :  2/5=4/10

If we do the cross multiplication, we will get

5*4 = 2 *10

20 = 20 so, : 2/5=4/10 is a proportion.

Ex  2:  2/5=4/10

Sol :    1/5=4/10

If we do the cross multiplication, we will get

5 X4 = 1 X 10

20 = 10 so, : 1/5=4/10 is not a proportion.

Geometry Math Project

Introduction to Geometry Math Project:

              A geometry math project is an interesting assignment (project) work which tends the student to work independently and let them analyze the mathematical observations to find the valuable solution which may be a new invention or change of the existing one. It improves the student’s knowledge in the Geometry and tends them to develop new formulas and make them solve the critical problems. Geometry is a division of Mathematics, which also suits everyone to face their life problems.
             
A Geometry Math Project

              Geometry is the major part of the mathematics that deals with the size, shape, relative position of figures, and the properties of space. There are many types of problems in geometry. When students involve in doing the geometry project, they have to choose the major topic and make themselves involved in the topic.

          
Steps to be followed to do a geometry math project

   To do a geometry math project, students have to follow the below steps.  
•    They have to choose the major subject from the geometry divisions in mathematics.
•    After the topic was chosen, the students have to start collecting the complete details of the subject.
•    Details may include the failures and the mistakes occurred during the time of calculation and they have to study about each sub topic to analyze the data.
•    After understanding about the topic, they have to do some experiments with the help of sample problems. Sample problems should cover all the formulas that have to be modified. 
•    In the experiment, they can use pictures, graphs or drawings to enhance their presentation.
•    Finally they have to give complete details about the topics like definition, properties, characteristics, etc.
•    By completing the experiments, students have to see the changes with the help of the existing project and have to prepare a report.

Completing the Geometry Math Project

              In any project, the final report takes a major part which helps them to complete the project in a successful way and they have to prepare a presentation on the project title. The final decision of the project may be applicable in real-time  situation.

Algebra made easy here !

Let's gain a quality learning of algebra through today's post specifically concerning with algebraic expressions.

Algebra is the study of expressions and variables. The main operation with algebra is finding the real numerical value of algebraic expressions. Usually algebraic expressions are represented as for example: 2x+3y = 0. Here, x and y are the unknown variables. These unknown variables are also termed as algebra symbols. Algebra symbols are very important in the study of algebra.

Other symbolic representation in algebra are: equivalence, approximate to, greater than and less than, parentheses, brackets, braces and so on.

For more help with this, connect with an online tutor and get your required help. You can connect with free algebra tutoring and learn the concept with better hand.

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Learn to mulitply fractions with me

Learn how to multiply fractions in today's post. Multiplying fractions is the easiest of all mathematical operations of fractions.

Multiplying fractions unlike adding and subtracting fractions with unlike denominators doesn't involve finding LCM. While multiplying fractions, the numerator should be multiplied by the numerator and the denominator should be multiplied by the denominator.

As for example: Multiply 2/3 * 5/6 = 10/18 = 5/9

For more help with this, one can avail to an online tutor and get the required help. Not just fractions help but one can opt for calculus tutoring and others as well.

Next time we will move on learning some other concept. Till then understand this concept and do practice problems to improve.