How to calculate the sum of squares?

The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five. 

A Number system or numeral system is defined as an elementary system to express numbers and figures. It is the unique way of representing of numbers in arithmetic and algebraic structure.

Numbers are used in various arithmetic values applicable to carry out various arithmetic operations like addition, subtraction, multiplication, etc which are applicable in daily lives for calculation. The value of a number is determined by the digit, its place value in the number, and the base of the number system. Numbers generally also known as numerals are the mathematical values used for counting, measurements, labeling, and measuring fundamental quantities.

Numbers are the mathematical values or figures used for the purpose of measuring or calculating quantities. It is represented by numerals as 2, 4, 7, etc. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.

What is the Sum of Squares?

The sum of the squares of numbers is referred to as the sum of squared values of the numbers. It's basically the addition of squared numbers.  

Here 2 terms, 3 terms, or ‘n’ number of terms, first n odd terms or even terms, set of natural numbers or consecutive numbers, etc. could be squared terms

We used to perform the arithmetic operation of the addition of squared numbers. Here we will come across the formula for the addition of squared terms.

• Sum of Squares Formula
• Sum N Terms of arithmetic series
• Sum Of Geometric progression
• Sum Of Nth Terms

In arithmetic, we come across the formula for the sum of n natural numbers. There are so many formulae and techniques for the calculation of the sum of squares. Let us use some of the formulae with respect to two numbers, three numbers, and n numbers. The square of a number is denoted by n².

a² + b² → Sum of two numbers a and b

a² + b² + c² → Sum of three numbers a, b and c

(a₁)² + (a₂)² + …. + (a_n)²→Sum of squares of n numbers

In terms of stats, this is equal to the sum of the squares of variation between individual values and the mean, i.e.,

Σ(a_i + \bar{a})^2

Where a_i represents individual values and \bar{a}  is the mean.

How to calculate the sum of squares?

Below are the formulas for sum of squares:

Formula 1: For addition of squares of any two numbers a and b is represented by:

a²+ b² = (a + b)² – 2ab

where a and b are real numbers

As per algebraic identities, we know;

(a + b)² = a² + b² + 2ab

Therefore, we can write the above equation as;

a²+b² = (a + b)² – 2ab

Formula 2: For squares of any three numbers say a, b, and c is represented by:

a² + b² + c² = (a + b + c)² - 2ab - 2bc - 2ac

where a, b and c are real numbers

From the algebraic identities, we know;

(a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ac

Therefore, we can write the above equation as;

a² + b² + c² = (a + b + c)² - 2ab - 2bc - 2ac

Formula 3: For nth Natural Numbers

The natural numbers include all the counting numbers, starting from 1 till infinity. If nth consecutive natural numbers are 1, 2, 3, 4, …, n, then the sum of squared ‘n’ consecutive natural numbers is represented by 1² + 2² + 3² + … + n².

it is also denoted by the notation Σn². The formula for the addition of squares of natural numbers is given below:

Σn² = [n(n + 1)(2n + 1)]/6

Formula 4: Sum of Squares of First n Odd Numbers

The addition of squares of first odd natural numbers is represented by:

Σ(2n - 1)² = [n(2n + 1)(2n - 1)]/3

Related Articles:

• Sum of Squares of 'n' Natural numbers
• Sum of Squares: Definition, Formula, Examples
• How to find the Sum and Difference of Squares?

Sample Questions - How to calculate the sum of squares?

Question 1: Evaluate 5² + 5² with the help of formula and directly as well. Verify the answers.

Solution: 

As We know;

a² + b² = (a + b)² - 2ab

5² + 5² = (5 + 5)² – 2×5×5

25 + 25 = 100 – 50

        50 = 50

Now, directly, we get;

5² + 5² = 25 + 25 = 50

Both answers are the same. Hence, verified.

Question 2: Find the addition of squares of the first 50 natural numbers.

Solution: 

Here The formula : sum of squared natural numbers is given by:

Σn² = [n(n + 1)(2n + 1)]/6

Here, n = 50

Σ50²  = (50/6) (50 + 1)(2 × 50 + 1)

Σ50²  = (25/3) (51)(101)

Σ50² = (25)(51)(37)

Σ50² = 47175

Question 3: Evaluate 4² + 4² + 4²  with the help of formula and directly as well. Verify the answers.

Solution: 

As we know 

a² + b² + c² = (a + b + c)² - 2ab - 2bc - 2ac

4² + 4² + 4² = (4 + 4 + 4)² - 2×4×4 - 2×4×4 - 2×4×4

16 + 16 + 16 = 144 - 32 - 32 - 32 

48 = 48

Now directly we get 

= 4² + 4² + 4²

= 16 + 16 + 16 

= 48 

Hence verified