Sum of Squares Formula

Introduction to the Sum of Squares Formula

Maths is all about numbers. Numbers are put into two categories, one is an odd number and the other one is an even number. The whole numbers that cannot be divided into pairs are known as odd numbers. When divided by 2 the odd numbers give a reminder ‘1’.The sum of two odd numbers gives an even number. The product of two or more than two odd numbers gives an odd number. The product of an even number and the product of an odd number is always an even number. In the number line, the first odd number is 1. The integer which is not an odd number is an even number. When an odd number is divided by two the result always comes out as a fraction.


Sum of Squares of n Natural Numbers Formula

  • Sum of Squares

The Sum of squares is the sum of the squares of numbers. Generally, it is the addition of the squared numbers. The squared terms can be of two terms, three terms, or even of ‘n’ terms. The first n even terms or the odd terms are the set of natural numbers or the consecutive numbers, etc. This is the basic math used to perform the arithmetic operation of the addition of the squared numbers.


In arithmetic operations, we often come across the sum of ‘n’ numbers. There are many formulas as well as techniques for the calculation of the sums of squares. In the statistics. It is always equal to the sum of the squares of the variation between the individual values and also the mean that is \[\Sigma(X_{i} + \overline{X})^{2}\] 

  • Sum of Squares Formula

The squares formula is always used to calculate the sum of two or more than two squares in an expression. To describe how well a model can represent the data being modeled the sum of squares formula is always used. The sum of the squares is the measure of the deviation from the mean value of the data. Therefore it is calculated as the total summation of the squares minus the mean.


The sum of the squares can be calculated using the formulas that are by the algebra and by the mean. The formula to calculate the sum of the squares of the two values are:


What is the sum of squares formula in statistics, algebra, and in 'n' terms?

  • In Statistics: Sum of Squares = \[\Sigma(X_{i} + \overline{X})^{2}\]  

  • In Algebra: Sum of Squares of Two Values = \[ a^{2} + b^{2} = (a + b)^{2} - 2ab \] 

  • For “n” Terms: Sum of Squares Formula for “n” numbers = \[= 1^{2} + 2^{2} + 3^{2}..........n^{2} = \frac{n(n+1)(2n+1)}{6} \]

Where,

  • ∑ = sum

  • \[X_{i}\] = each value in the set 

  • \[\overline{X}\] = mean

  • \[X_{i} - \overline{X}\] = deviation

  • \[ (X_{i} - \overline{X})^{2}\] = square of the deviation

  • a, b = numbers

  • n = number of terms


Calculating the sum of squares of the data has many applications in real life. In statistics, the sum of squares measures how far individual measurements are from the mean. This formula is used to describe how well a model represents the data being modeled and it also gives the measure of deviation from the mean value. This is why the formula is calculated as the subtraction of the total summation of the squares and the mean. It is a very useful tool to evaluate the overall variance of a data set from the mean value. It is very useful in many situations for example, the instability in the cardiovascular system that requires medical attention. Statisticians and scientists use this tool. 


In Statistics

Sum of Squares:

\[ = \Sigma (X_{i} + \overline{}) ^{2} \] 

In Algebra

Sum of Squares of Two Values:

\[ a^{2} + b^{2} = (a + b)^{2} - 2ab \]

For “n” Terms

Sum of Squares Formula for “n” numbers

\[= 1^{2} + 2^{2} + 3^{2}..........n^{2} = \frac{n(n+1)(2n+1)}{6} \]


An example of the sum of squares formula

Example 1: Find the value of 42 + 62 using the sum of squares formula?

Solution:

To find: 42 + 62


Given: a = 4, b = 6 


Using the sum of squares formula,

 a2 + b2 = (a + b)2 − 2ab


42 + 62 = (4 + 6)2 − 2(4)(6)


= 100 − 2(24)


= 100 − 48


= 52


Answer: The value of 42 + 62 is 52.

 

Example 2: Sum up the following series. 12 + 22 + 32 ……. 1002


Solution:


Find the sum of the series


Using the sum of squares formula, find the sum of the series.


Sum of Squares \[= 1^{2} + 2^{2} + 3^{2}..........n^{2} = \frac{n(n+1)(2n+1)}{6} \]


Given: n = 100 


= [100(100+1)(2×100+1)]/6


= [100(101)(201)]/6


= 338350.


Answer: Sum of the given series is 338350.

 

Example 3: Determine the sum of the squares of 10 and 22 directly using the formula. Verify the answers.


Solution: 


102 + 222 = 100 + 484 = 584


Using the formula a2 + b2 = (a + b)2 - 2ab, we get 102 + 222 = (10 + 22)2 - 2 × 10 × 22 


= 322 - 440


= 1024 - 440 = 584, Thus verified.

FAQs on Sum of Squares Formula

1. Algebra: What Is the Sum of Squares Formula?

As one of the most important algebraic identities, the sum of squares formula is important. The formula is a2 + b2 and is read as "a squared plus b square". Sum of squares (a2 + b2) formula is written as a2 + b2 = (a + b)2 -2ab


Using the following example, let us learn how to use the sum of squares formula, i.e., a2 + b2.

2. Using the sum of squares formula, find the value of 102 + 22.

To find: 102 + 22

Let us assume, a = 10 and b = 2.


By substituting a2 + b2 in the sum of squares formula, we will arrive at the sum of squares formula. 


a2 + b2 = (a +b)2 -2ab


102 + 22 = (10+2)2 - 2(10)(2)


= 144 - 40 = 104


102 + 22 = 104.

3. What are the steps for using the sum of squares formula?

In order to use the sum of squares formula, the following steps need to be followed.

  • Let's first observe the pattern of two numbers, whether the numbers have the power of two or not, in the form of a2 + b2.

  • Use the sum of squares formula a2 + b2 = (a + b)2 -2ab 

  • Simply substitute the values of a and b in the sum of squares a2 + b2 formula.

4. What is the Expansion of the Sum of Squares Formula?

The a2 + b2 formula is also known as the square sum formula and is written as a square plus a square. This expression is written as a2 + b2 = (a + b)2 -2ab. 

5. What is the Difference Between the Sum of Squares of First n Even Numbers and Odd Numbers?

The addition of squares of the first even natural numbers:

Σ(2n)2 = 22 + 42 + 62 + 82 + …+ (2n)2

While the addition of squares of the first odd natural numbers

Σ(2n-1)2 = 12 + 32 + 52 + … + (2n – 1)2

6. Why is it Important to Learn Formulas and Equations?

Computer chips used in all the machines we use in our daily routine like washers, dryers, backs, etc. All the chips that we use in these machines are based on mathematical equations, formulas, and algorithms.


Mathematical equations and formulas are also used in traffic control, aircraft, space programs and medicine, etc.

7. When was Formula Invented in Mathematics?

The first formula was invented by the Babylonians and the derivation was of the square root of 2. About 1800 B.C .An Egyptian papyrus contains the first ever quadratic solution and about 1650 B.C. Another papyrus contains solving linear equations and also the first example of cotangent.

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