Sum of Squares

Sum of squares refers to the sum of the squares of numbers. It is basically the addition of squared numbers. The squared terms could be 2 terms, 3 terms, ‘n’ number of terms, first n even terms or odd terms, set of natural numbers or consecutive numbers, etc. This is basic math, used to perform the arithmetic operation of addition of squared numbers. In this article, we will come across the formula for addition of squared terms with respect to statistics, algebra, and for n number of terms.

Definition

In arithmetic, we often come across the sum of n natural numbers. There are various formulae and techniques for the calculation of the sum of squares. Let us write some of the forms with respect to two numbers, three numbers and n numbers.

  • x2 + y2 → Sum of two numbers x and y
  • x2+y2+z2 → Sum of three numbers x, y and z
  • (x1)2+(x2)2+….+(xn)2→Sum of squares of n numbers

In statistics, it is equal to the sum of the squares of variation between individual values and the mean, i.e.,

Σ(xi + x̄)2

Where xi represents individual values and x̄ is the mean.

Sum of Squares Formulas and Proofs

  1. For Two Numbers:

The formula for addition of squares of any two numbers x and y is represented by;

x2 + y2 = (x + y)2– 2ab ; x and y are real numbers

Proof: From the algebraic Identities, we know;

(x + y)2 = x2 + y2 + 2ab

Therefore, we can write the above equation as;

x2+y2 = (x + y)2 – 2ab

  1. For Three Numbers

The formula for addition of squares of any three numbers say x, y and z is represented by;

x2 + y2+z2 = (x+y+z)2-2xy-2yz-2xz ; x,y and z are real numbers

Proof: From the algebraic Identities, we know;

(x+y+z)2 = x2 + y2 + z2 + 2xy + 2yz + 2xz

Therefore, we can write the above equation as;

x2 + y2+z2 = (x+y+z)2-2xy-2yz-2xz

  1. For n Natural Numbers

The family of natural numbers includes all the counting numbers, starting from 1 till infinity. If n 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.

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

Σn2 = n/6(n +1)(2n + 1)

Sum of Squares of First n Even Numbers

The addition of squares of first even natural numbers is given by:

Σ(2n)2 = 22 + 42 + 62 + 82 + …+ (2n)2

Proof:

Σ(2n)2 = 22.12 + 22.22 + 22.32 + 22.42 +…+ 22.n2

Σ(2n)2 = 22(12 + 22 + 32 + 42 … + n2)

Σ(2n)2 = 4[n/6(n +1)(2n + 1)] (Formula for sum of squared n natural numbers)

Σ(2n)2 = 2n/3 (n+1) (2n+1)

Sum of Squares of First n Odd Numbers

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

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

Proof:

Σ(2n-1)2 = 12 + 22 + 32 + … + (2n – 1)2 + (2n)2 – [22 + 42 + 62 + … + (2n)2]

On applying the formula for the addition of squares of 2n natural numbers and of n even natural numbers, we get;

Σ(2n-1)2 = 2n/6 (2n + 1)(4n + 1) – (2n/3) (n+1)(2n+1)

Σ(2n-1)2 = n/3 [(2n+1)(4n+1)] – 2n/3 [(n+1)(2n+1)]

Taking out the common terms, we get;

Σ(2n-1)2 = n/3 (2n+1) [4n + 1 – 2n -2]

Σ(2n-1)2 = n/3 (2n+1) (2n -1) is the required expression.

Sum of:

Formula

Squares of two numbers

x2 + y2 = (x+y)2-2ab

Squares of three numbers

x2 + y2+z2 = (x+y+z)2-2xy-2yz-2xz

Squares of first ‘n’ natural numbers

Σn2 = n/6(n +1)(2n + 1)

Squares of first even natural numbers

Σ(2n)2 = 2n/3 (n+1) (2n+1)

Squares of first odd natural numbers

Σ(2n-1)2 = n/3 (2n+1) (2n -1)

Sum of Squares Examples

Q.1: Evaluate 42 + 52 by the help of formula and directly as well. Verify the answers.

Solution: We know;

x2 + y2 = (x+y)2-2ab

42 + 52 = (4 + 5)2 – 2.4.5

42 + 52 = 92 – 40

42 + 52 = 81 – 40 = 41

Now, solving the given equation directly, we get;

42 + 52 = 16 + 25 = 41

Both answers are the same. Hence, verified.

Q.2: Find the addition of squares of the first 40 natural numbers.

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

Σn2 = n/6(n +1)(2n + 1)

Here, n = 40

Σ402 = 40/6 (40 + 1)(2 x 40 + 1)

Σ402 = 20/3 (41)(81)

Σ402 = (20)(41)(27)

Σ402 = 22140