Bradley completed a proof of his conjecture but before he could hand it in to his teacher he dripped tomato sauce on it. He is rewriting his proof but there are places where the tomato sauce has covered his work. help Bradley fill in the gaps of his proof.
Let 2a represent an ___ integer and 2b + 1 represent and odd integer.
(2a)^2 + (2b+1)^2 = 4a^2 + (2b + 1) (2b + 1)
= 4a^2 + 4b^2 + 2b + 2b +1
= 4a^2 + 4b^2 + 4b + 1
= 2(2a^2 + 2b^2 +2b) + 1
This is an odd integer. It is in the form of 2 times a number plus 1.
- Explain how Bradley uses both inductive and deductive reasoning in his process.
Bradley uses inductive reasoning in his process by observing patterns and making a conjecture. Initially, he notices that when he squares an even integer (represented by 2a), the result is always a multiple of 4. When he squares an odd integer (represented by 2b+1), he notices that the resulting number is always one more than a multiple of 4. Based on these patterns, Bradley makes the conjecture that the sum of the squares of an even and an odd integer will always result in an odd integer.
Bradley then uses deductive reasoning to prove his conjecture. He starts by substituting 2a for the even integer and 2b+1 for the odd integer in the equation. He then simplifies the equation step by step, using logical deductions and mathematical operations. Finally, he reaches the conclusion that the sum of the squares of an even and an odd integer is in the form of 2 times a number plus 1, which is an odd integer. This deductive reasoning process allows him to logically prove his conjecture.
Bradley completed a proof of his conjecture but before he could hand it in to his teacher he dripped tomato sauce on it. He is rewriting his proof but there are places where the tomato sauce has covered his work. help Bradley fill in the gaps of his proof.
Let 2a represent an ___ integer and 2b + 1 represent and odd integer.
(2a)^2 + (2b+1)^2 = 4a^2 + (2b + 1) (2b + 1)
= 4a^2 + 4b^2 + 2b + 2b +1
= 4a^2 + 4b^2 + 4b + 1
= 2(2a^2 + 2b^2 +2b) + 1
This is an odd integer. It is in the form of 2 times a number plus 1.
- Explain how Bradley uses both inductive and deductive reasoning in his process.
In three sentence
Bradley uses inductive reasoning by observing patterns and making a conjecture based on those patterns, such as recognizing that squaring an even integer always results in a multiple of 4. Then, he uses deductive reasoning to prove his conjecture by substituting variables and applying logical deductions step by step, such as in the simplification of the equation. This combination of inductive and deductive reasoning allows Bradley to make an initial conjecture and then logically prove it through deductive reasoning.
Bradley completed a proof of his conjecture but before he could hand it in to his teacher he dripped tomato sauce on it. He is rewriting his proof but there are places where the tomato sauce has covered his work. help Bradley fill in the gaps of his proof.
Let 2a represent an ___ integer and 2b + 1 represent and odd integer.
(2a)^2 + (2b+1)^2 = 4a^2 + (2b + 1) (2b + 1)
= 4a^2 + 4b^2 + 2b + 2b +1
= 4a^2 + 4b^2 + 4b + 1
= 2(2a^2 + 2b^2 +2b) + 1
This is an odd integer. It is in the form of 2 times a number plus 1.
- Explain how Bradley uses both inductive and deductive reasoning in his process.
In two sentence
Bradley uses inductive reasoning by observing patterns and making a conjecture based on those patterns, such as recognizing the pattern of the sum of squares resulting in an odd integer. Then, he uses deductive reasoning to prove his conjecture by logically analyzing and manipulating the given equation to arrive at a conclusion that aligns with his initial observation.
Bradley uses both inductive and deductive reasoning in his process as follows:
1. Inductive Reasoning: Bradley starts by making observations based on specific cases. In this case, he considers the expressions (2a)^2 and (2b+1)^2, and he notices that when he expands these expressions, they both have the form of a squared term plus a constant term. He observes this pattern and generalizes it for any values of a and b.
2. Deductive Reasoning: Bradley then uses deductive reasoning to prove his conjecture based on the patterns he observed through inductive reasoning. He takes the generalized form he derived and manipulates the algebraic expressions step-by-step to arrive at a conclusion. Each step in the proof is based on deductive reasoning, applying logical operations and properties of numbers.
Through this combination of inductive and deductive reasoning, Bradley is able to formulate and prove his conjecture.
Bradley uses both inductive and deductive reasoning in his process of filling in the gaps of his proof.
Inductive Reasoning:
1. Bradley starts with a few specific cases to observe patterns or relationships. In this case, he uses the representation of even and odd numbers: 2a for even integers and 2b + 1 for odd integers. By analyzing the sum of their squares, he observes that the result is always an odd integer.
Deductive Reasoning:
1. Once Bradley notices the pattern, he uses deductive reasoning to prove his conjecture. He proceeds by manipulating the given expressions algebraically to arrive at a general formula. He squares 2a and (2b + 1), expands the expression, and simplifies it step by step using algebraic operations.
2. Through algebraic manipulation, Bradley finally arrives at the expression: 2(2a^2 + 2b^2 + 2b) + 1, which shows that the result is in the form of 2 times a number plus 1, indicating an odd integer.
By using inductive reasoning to observe a pattern and then applying deductive reasoning to prove that pattern, Bradley fills in the gaps of his proof and confirms his conjecture.