[Maths Class Notes] on Sum of Squares Pdf for Exam

The notes available at .com for the concept of sum of squares ensure that your success is guaranteed. In these notes, students get to learn about the procedure of calculating the sum of squares of two numbers, three numbers, natural numbers, Even numbers and odd numbers and their formulae. This topic is a necessity for the students who wish to learn Maths and practice questions to make themselves better at calculations. You will also find some interesting fun facts to refresh your studies and clear your doubts about the concept of sum of squares. To understand this chapter, every student must first revise the basics such as what a square of a number is and how it is calculated. The student who does this is sure to understand every single point made in these notes effectively.

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Sum of Squares of Numbers:

A Square of a Number is the product obtained when the Number is multiplied by itself. Square Numbers are those Numbers whose Square root is an integer. If ‘a’ is a Number, then its Square is obtained by multiplying ‘a’ by itself i.e. a x a. The Square of a Number ‘a’ is represented as a2 and is read as ‘a’ Squared. Sum of the Squares is obtained by adding the Squares of the Numbers. This section elaborates a detailed explanatory note on standard formulae for Sum of Squares of Natural Numbers, Sum of Squares of any two or three Numbers. 

Sum of Squares Formula for Two Numbers:

Consider any two Numbers ‘R’ and ‘S’. The Sum of the Numbers is written as ‘R + S’ and their difference is written as ‘R – S’. The Sum of Squares of these two Numbers can be calculated under two different cases.

Case 1: Squaring the Sum of the Numbers:

Sum of the two Numbers ‘R’ and ‘S’ is written as ‘R + S’. Its Square is calculated as:

(R + S)2 = (R + S) (R + S)

= R. R + R. S + S. R + S. S

= R2 + RS + RS + S2

(R + S)2 = R2 + 2 RS + S2

Transfering 2 RS to the other side of the equation, we get

i.e. The Sum of Squares of two Numbers is equal to the difference obtained when twice the product of the Numbers is subtracted from the Square of their Sum.

Case 2:  Squaring the Difference Between the Numbers:

The difference between the two Numbers ‘R’ and ‘S’ is written as ‘R – S’. Its Square is calculated as:

(R – S)2 = (R – S) (R – S)

= R. R – R. S – S. R + (-S). (-S)

= R2 – RS – RS + S2

(R – S)2 = R2 – 2 RS + S2

Transfering – 2 RS to the other side of the equation, we get

i.e. Sum of Squares of two Numbers can also be calculated as the Sum of the Squares of their difference and twice their product. 

Sum of Squares Formula for Three Numbers:

Sum of three Numbers ‘L’, ‘M’ and ‘N’ is written as L + M + N. Its Square is calculated as follows:

(L + M + N)2 = (L + M + N) (L + M + N)

= L2 + LM + LN + ML + M2 + MN + NL + NM + N2

= L2 + M2 + N2 + 2 LM + 2 MN + 2 NL

(L + M + N)2 = L2 + M2 + N2 + 2 (LM + MN + NL)

The above formula gives the Sum of Squares of three Numbers.

The Sum of Squares of Natural Numbers formula is given as

Sum of Squares of First n Natural Numbers Formula:

The Sum of the Squares of first n Natural Numbers is calculated as the quotient obtained when the product of the Number of Natural Numbers whose Sum of the Squares is to be determined, one more than the Number of Natural Numbers and one more than twice the Number of Natural Numbers is divided by the positive integer 6.

Sum of Squares of Even Numbers Formula:

An Even Number is generally represented as a multiple of 2. Let us con
sider an Even Number ‘2p’. The Sum of Squares of Even Numbers is calculated by substituting 2p in the place of ‘p’ in the formula for finding the Sum of Squares of first n Natural Numbers. In this case n = p.

Sum of Squares of Natural Numbers formula is 

[sum p^{2} = frac{p(p+1)(2p+1)}{6}]

Substituting p = 2p in the above equation, we get

[sum (2p)^{2} = sum 2^{2} . p^{2} = 4sum p^{2}]

[sum (2p)^{2} = 4(frac{n(n+1)(2n+1)}{6})]

[sum (2p)^{2} = frac{2n(n+1)(2n+1)}{3}]

Sum of Squares of Odd Numbers Formula:

The general representation of an Odd Number is given as ‘(2p – 1)’. The Sum of Squares of Odd Numbers can be calculated as:

[sum (2p – 1)^{2} = frac{p(2p+1)(2p-1)}{3}]

Fun Facts:

• A Square Number always ends with 0, 1, 4, 5, 6, and 9. Any Number with digits 2, 3, 7, and 8 are not Square Numbers.

• All counting Numbers are called Natural Numbers. 

The Numbers that are completely divisible by two are called Even Numbers. The Numbers that have ‘1’ as a remainder when divided by two are called Odd Numbers.