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

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.