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, the 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 the purpose of 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 are 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 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 + …. + (an)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.,
[Tex]Σ(a_i + \bar{a})^2
[/Tex]
Where ai represents individual values and [Tex]\bar{a}
[/Tex] is the mean.
Formulae 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
Sample Questions
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
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