Maths-
General
Easy
Question
Is the given conjecture correct? Provide arguments to support your answer.
The product of any two consecutive odd 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: odd numbers.
Let the odd number be 2n+1. Its consecutive even number will be 2n+3
Finding the product of (2n+1) and (2n+3)
(2n+1)(2n+3) = 2n(2n+3) + 1(2n+3)
(2n+1)(2n+3) = 2n(2n) + 2n(3) + 1(2n) + 1(3)
(2n+1)(2n+3) = 4n2 + 6n + 2n + 3
(2n+1)(2n+3) = 4n2 + 8n + 3
The given conjecture is “The product of any two consecutive odd numbers is 1 less than a perfect square”
So, add 1 to 4n2 + 8n + 3
4n2 + 8n + 4 = (2n)2 + 2(2n)(2) + 22
= (2n + 2)2
So, we can see that the perfect square of 2n+2 is 4n2 + 8n + 4 and 1 less than 4n2 + 8n + 4 is a product of any two consecutive odd numbers.
Final Answer:
Hence, the given conjecture is right and we have proved it by Deductive reasoning.
The given conjecture is “The product of any two consecutive odd numbers is 1 less than a perfect square”
So, add 1 to 4n2 + 8n + 3
So, we can see that the perfect square of 2n+2 is 4n2 + 8n + 4 and 1 less than 4n2 + 8n + 4 is a product of any two consecutive odd numbers.
Final Answer:
Hence, the given conjecture is right and we have proved it by Deductive reasoning.
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