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Sum of Squares

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Last updated date: 05th Mar 2024
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Notes on Sum of Squares

The notes available at Vedantu.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.


To avail study material of subjects Mathematics or other than Maths  like English and science for students of CBSE as well as ICSE board the students can visit Vedantu.com and sign in to enjoy free study material at your screen for absolutely free. 


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 of two Numbers

g2 + h2 = (g + h)2 - 2 gh

Sum of Squares of three Numbers

f2 + g2 + h2 = (f + g + h)2 - 2 (fg + gh + fh)

Sum of Squares of Natural Numbers formula

n(n+1)(2n+1)6

n(n+1)(2n+1)6

Sum of Squares of Even Numbers

2n(n+1)(2n+1)3

2n(n+1)(2n+1)3

Sum of Squares of Odd Numbers

n(2n+1)(2n−1)3

n(2n+1)(2n−1)3


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

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


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

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


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)

L2 + M2 + N2 = (L + M + N)2 - 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:

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


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 consider 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

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

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

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


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.

FAQs on Sum of Squares

1. How do we Calculate the Sum of Squares of the First 100 Natural Numbers?

The Sum of the Squares of first n Natural Numbers is calculated by a standard formula in Mathematics. The Sum of Squares of Natural Numbers formula is given as:

∑p² = p(p+1)(2p+1)/6


In the above formula, ‘p’ is the Number of Natural Numbers whose Sum is to be calculated and the symbol Σ indicates the Sum of the Squares of ‘p’ Natural Numbers.

 

If we have to calculate the Sum of Squares of the first 100 Natural Numbers, the value of ‘p’ should be taken as 100. So, the Sum of Squares of the first 100 Natural Numbers is calculated as:  

∑p² = 100(100+1)(2(100+1))/6

∑p² = 100(101)(201)/6

∑p² = 2030100/6 = 338350

So, the Sum of Squares of first n Natural Numbers where n = 100 is 338350.

2. How is the Sum of Squares of two Numbers Calculated when the Sum and Difference Between the Numbers are Given?

Consider two Numbers ‘m’ and ‘n’. Sum of the Squares of these two Numbers can be calculated in two different ways.

Case 1: Considering the Square of Sum of the Numbers

According to algebraic identities, it is evident that the Square of the Sum of any two Numbers is equal to the Sum of the Squares of the two Numbers added to twice the product of the Numbers. So, the Sum of the Squares of two Numbers can, therefore, be calculated as the difference obtained when twice the product of the Numbers is subtracted from the Square of their Sum. 

m2 + n2 = (m + n)2 - 2 mn


Case 2: Considering the Square of difference between the Numbers

The algebraic identity to find the Square of difference between two Numbers is as follows: The Square of the difference between two Numbers is equal to the difference obtained when twice the product of the two Numbers is subtracted from the Sum of their Squares. By slight alterations in this formula, the Sum of Squares of two Numbers can be calculated as the Sum of the Square of the difference between the two Numbers and twice their product. 

m2 + n2 = (m - n)2 + 2 mn

3. Where can I get the ‘Sum of Squares’ notes?

The students can get everything regarding Mathematics at Vedantu.com. You can find study material which is available in PDF form as well. All the students have to do is search the notes for ‘Sum of Squares’ with Vedantu in the title or just easily sign in using the Vedantu website. These study materials are curated by master teachers so every student understands each topic without any problem. The notes can be easily copied from the website for the convenience of students to make their studies easy. 

4. How can I understand the ‘Sum of Squares’?

The students can understand the concept of the Sum of Squares if they are already introduced to the concept of what a Square of a Number means. The Square of a Number means a Number is multiplied by itself. For example, the Square of 2 will be 2 × 2= 2²= 4. Or another example, the Square of 4 will be 4 × 4 = 4²= 16. The Sum of Squares of different Numbers whether they are Odd or Even or just two Natural Numbers can be calculated by studying the concept of ‘Sum of Squares’ available at Vedantu.com for absolutely free of charge.

5. Why is it important to understand ‘Sum of Squares’?

The students who are currently studying Maths in classes such as 6,7, 8, 9 or 10 need to know about the concept of ‘Sum of Squares’ as it will help in their Maths examinations of the aforementioned classes. It is also important for those students who wish to study Maths in higher classes as the concept of Sum of Squares is used in many questions and to calculate this without any hassle is something every Maths scholar must know. This concept is useful for competitive exams too as the calculations become easier to deal with if you are well versed with this concept enough.