Maths-

General

Easy

Question

# Is the given conjecture correct? Provide arguments to support your answer.

The product of any two consecutive even numbers is 1 less than a perfect square

Hint:

### Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on the observations and this conclusion is called conjecture.

Deductive reasoning is the process by which a person makes conclusions based on previously known facts.

## The correct answer is: even numbers.

### Let the even number be 2n. Its consecutive even number will be 2n+2

Finding the product of 2n and (2n+2)

2n(2n+2) = 2n(2n) + 2n(2)

2n(2n+2) = 4n^{2} + 4n

The given conjecture is “The product of any two consecutive even numbers is 1 less than a perfect square”

So, add 1 to 4n^{2} + 4n

4n^{2} + 4n +1 = (2n)^{2} + 2(2n)(1) + 1^{2}

= (2n + 1)^{2}

So, we can see that the perfect square of 2n+1 is 4n^{2} + 4n +1 and 1 less than 4n^{2} + 4n +1 is a product of any two consecutive even numbers.

Final Answer:

Hence, the given conjecture is right and we have proved it by Deductive reasoning.

^{2}+ 4n

The given conjecture is “The product of any two consecutive even numbers is 1 less than a perfect square”

^{2}+ 4n

^{2}+ 4n +1 = (2n)

^{2}+ 2(2n)(1) + 1

^{2}

^{2}

So, we can see that the perfect square of 2n+1 is 4n

^{2}+ 4n +1 and 1 less than 4n

^{2}+ 4n +1 is a product of any two consecutive even numbers.

Final Answer:

Hence, the given conjecture is right and we have proved it by Deductive reasoning.

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