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It seems that you uploaded the same image again. As mentioned earlier, the task is to use "Definition 1.6" to explain why \( 2n + 7 \) 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 \( n \) is even if there exists some integer \( k \) such that \( n = 2k \).
   
2. **Odd integer**: An integer \( n \) is odd if there exists some integer \( k \) such that \( n = 2k + 1 \).

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

Thus, \( 2n + 7 \) is odd for any integer \( n \).

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 \( n = 0 \), what is \( 2n + 7 \)?
4. Why is the sum of two even numbers always even?
5. Is \( 2n + 5 \) odd or even for an integer \( n \)?

### 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.

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

Use algebraic reasoning to explain why 2n + 7 is an odd integer. The problem explores the concepts of even and odd integers and is suitable for students in Grades 8-10.

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