Question

# Make and test a conjecture about the square of odd numbers.

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: Hence, we conclude that the square of an odd number will also be an odd number and we have proved this conjecture by deductive reasoning.

### Let’s create a conjecture for the odd numbers 3, 7, 9, and 15. We are asked to make a conjecture about the square of odd numbers.

So first, do the square of the assumed odd numbers. We got

3^{2} = 9

7^{2} = 49

9^{2} = 81

15^{2} = 225

So, we conclude that the square of an odd number will also be an odd number and this is the conjecture.

Now, let’s see if the conjecture is true or not by deductive reasoning. Let’s say the odd number is x. This odd can be written as

x = 2k + 1 (where k is some integer)

Now, squaring both sides

x^{2} = (2k+1)^{2}

= 4k^{2} + 1 + 4k

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

Take (k^{2}+k) = m, where m is some integer

x^{2} = 4m +1

4m is an even number and adding 1 to 4m will result in an odd number and hence x^{2} is an odd number.

Final Answer:

Hence, we conclude that the square of an odd number will also be an odd number and we have proved this conjecture by deductive reasoning.

^{2}= 9

^{2}= 49

^{2}= 81

^{2}= 225

So, we conclude that the square of an odd number will also be an odd number and this is the conjecture.

Now, let’s see if the conjecture is true or not by deductive reasoning. Let’s say the odd number is x. This odd can be written as

x = 2k + 1 (where k is some integer)

Now, squaring both sides

^{2}= (2k+1)

^{2}

^{2}+ 1 + 4k

^{2}+k) +1

Take (k

^{2}+k) = m, where m is some integer

^{2}= 4m +1

4m is an even number and adding 1 to 4m will result in an odd number and hence x

^{2}is an odd number.

Final Answer:

Hence, we conclude that the square of an odd number will also be an odd number and we have proved this conjecture by deductive reasoning.

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2. Unknown variables, numerical values, and arithmetic operators make up an algebraic expression. There are no symbols for equality or inequality in it.

¶In contrast to equations, the equal (=) operator is used between two mathematical expressions. Expressions denote a combination of numbers, variables, and operation symbols. The "equal to" sign also has the same value on both sides.

### Two rival dry cleaners both advertise their prices. Let x equal the number of items dry cleaned. Store A’s prices are represented by the equation 15x - 2. Store B’s prices are represented by the expression 3 (5x + 7). When do the two stores charge the same rate ? Explain.

Mathematical expressions are made up of at least two numbers or variables, one math operation, and a sentence. This mathematical operation allows you to multiply, divide, add, or subtract numbers.

¶**Types of Expression**

1. Arithmetic operators and numbers make up a mathematical numerical expression. There are no symbols for undefined variables, equality, or inequality.

2. Unknown variables, numerical values, and arithmetic operators make up an algebraic expression. There are no symbols for equality or inequality in it.

¶In contrast to equations, the equal (=) operator is used between two mathematical expressions. Expressions denote a combination of numbers, variables, and operation symbols. The "equal to" sign also has the same value on both sides.