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) = 4n2 + 4n
    The given conjecture is “The product of any two consecutive even numbers is 1 less than a perfect square”

    So, add 1 to 4n2 + 4n

    4n2 + 4n +1 = (2n)2 + 2(2n)(1) + 12

    = (2n + 1)2
    So, we can see that the perfect square of 2n+1 is 4n2 + 4n +1 and 1 less than 4n2 + 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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