Direct Math Proof: If n is odd then 3n + 7 is even

TL;DR

This video explains how to prove that if n is an odd integer, then the expression 3m + 7 is even.

Transcript

hi in this video we're going to do a proof the question is to prove that if n is an odd integer then 3m plus 7 is even before we do this proof we need some definitions so recall that we say that x is even and this is equivalent to saying that x is a multiple of two so basically you can write this as x equals 2k for some integer k so that's the defi... Read More

Key Insights

• #️⃣ Even numbers are multiples of 2, and odd numbers are one more than multiples of 2.
• 🛀 A direct proof assumes the hypothesis to be true and shows that the conclusion is also true.
• 😑 Substituting the expression for n in the given equation allows for manipulation and simplification.
• 👍 To prove that 3m + 7 is even, we show that it can be written as 2 times an integer (3k + 5).
• 😑 The importance of specifying that the resulting expression is an integer is highlighted in the proof.
• 🍹 Products and sums of integers result in integers.
• 😀 The proof concludes that 3m + 7 is an even integer, fulfilling the initial hypothesis.

Questions & Answers

Q: What are the definitions of even and odd numbers?

An even number is a multiple of 2 and can be written as 2k, where k is an integer. An odd number is one more than a multiple of 2 and can be written as 2k + 1, where k is an integer.

Q: What does a direct proof involve?

A direct proof assumes the hypothesis to be true and then demonstrates that the conclusion is also true. It starts by stating the hypothesis, defining any relevant terms, performing calculations, and arriving at the desired conclusion.

Q: How is the hypothesis expressed in the given proof?

The hypothesis is stated as "Suppose that n is an odd integer," which means that we are assuming n can be expressed as 2k + 1.

Q: How does the proof show that 3m + 7 is even?

By substituting the expression for n (2k + 1) into 3m + 7, performing the necessary calculations, and simplifying the expression, the proof demonstrates that 3m + 7 can be written as 2 times the integer 3k + 5, proving it to be even.

Summary & Key Takeaways

• The video introduces the definitions of even and odd numbers, stating that an even number can be expressed as 2k, and an odd number can be expressed as 2k + 1.

• The proof starts by assuming that n is an odd integer (n = 2k + 1) and aims to show that 3m + 7 is even.

• The proof replaces n with its odd integer representation, performs some calculations, and demonstrates that 3m + 7 can be expressed as 2 times an integer, proving it to be even.