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) = 4n^{2} + 6n + 2n + 3

(2n+1)(2n+3) = 4n^{2} + 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 4n^{2} + 8n + 3

4n^{2} + 8n + 4 = (2n)^{2} + 2(2n)(2) + 2^{2}

= (2n + 2)^{2}

So, we can see that the perfect square of 2n+2 is 4n^{2} + 8n + 4 and 1 less than 4n^{2} + 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.

^{2}+ 6n + 2n + 3

^{2}+ 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 4n

^{2}+ 8n + 3

^{2}+ 8n + 4 = (2n)

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

^{2}

^{2}

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

^{2}+ 8n + 4 and 1 less than 4n

^{2}+ 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.

### Related Questions to study

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### In an academic contest correct answers earn 12 points and incorrect answers lose 5

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ii)How many answers did each school get correct in the final round?

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¶The standard form of a linear equation in one variable is: ax + b = 0

Where,

The letters 'a' and 'b' are real numbers.

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### In an academic contest correct answers earn 12 points and incorrect answers lose 5

points. In the final round, school A starts with 165 points and gives the same number

of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.

ii)How many answers did each school get correct in the final round?

A linear equation in one variable is an equation that has only one solution and is expressed in the form ax+b = 0, where a and b are two integers and x is a variable. 2x+3=8, for example, is a linear equation with a single variable. As a result, this equation has only one solution, x = 5/2.

¶The standard form of a linear equation in one variable is: ax + b = 0

Where,

The letters 'a' and 'b' are real numbers.

'a' and 'b' are both greater than zero.

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### Find the value of 𝑚 & 𝑛 to make a true statement.

(𝑚𝑥 + 𝑛𝑦)^{2} = 4𝑥^{2} + 12𝑥𝑦 + 9𝑦^{2}

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### 37. In an academic contest correct answers earn 12 points and incorrect answers lose 5

points. In the final round, school A starts with 165 points and gives the same number

of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.

i)Which equation models the scoring in the final round and the outcome of the contest

**Different Types of Variables**

### 37. In an academic contest correct answers earn 12 points and incorrect answers lose 5

points. In the final round, school A starts with 165 points and gives the same number

of correct and incorrect answers. School B starts with 65 points and gives no incorrect answers and the same number of correct answers as school A. The game ends with the two schools tied.

i)Which equation models the scoring in the final round and the outcome of the contest

**Different Types of Variables**