Math Problem Statement

Let n be an integer. Use Definition 1.6 to explain why 2n + 7 is an odd integer.

Solution

It seems that you uploaded the same image again. As mentioned earlier, the task is to use "Definition 1.6" to explain why 2n+72n + 7 is an odd integer. Since I do not have access to the specific content of "Definition 1.6," I can only infer based on standard mathematical definitions.

If "Definition 1.6" is the standard definition of odd and even integers, we can proceed as follows:

Standard Definitions:

  1. Even integer: An integer nn is even if there exists some integer kk such that n=2kn = 2k.

  2. Odd integer: An integer nn is odd if there exists some integer kk such that n=2k+1n = 2k + 1.

Step-by-Step Explanation for 2n+72n + 7:

  • Let nn be any integer.
  • 2n2n is clearly an even integer because multiplying any integer by 2 results in an even number.
  • Now, adding 7 (which is odd) to 2n2n gives: 2n+7=even integer+odd integer.2n + 7 = \text{even integer} + \text{odd integer}. It is a known fact that the sum of an even integer and an odd integer is always an odd integer.

Thus, 2n+72n + 7 is odd for any integer nn.

If you want clarification on any part of the solution or details on "Definition 1.6" (if you can share more context), feel free to ask!

Related Questions:

  1. Can you show an example of an even integer and an odd integer sum?
  2. How do you formally prove that the sum of an even and odd integer is odd?
  3. If n=0n = 0, what is 2n+72n + 7?
  4. Why is the sum of two even numbers always even?
  5. Is 2n+52n + 5 odd or even for an integer nn?

Tip:

When handling proofs in mathematics, always start by writing down the key definitions and properties, as they often guide your steps clearly through logical reasoning.

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Math Problem Analysis

Mathematical Concepts

Even and Odd Integers
Basic Algebra

Formulas

n is even if n = 2k
n is odd if n = 2k + 1

Theorems

Sum of an even and odd integer results in an odd integer

Suitable Grade Level

Grades 8-10