Question
Show that the sum of two consecutiv odd numbers is an even number.
Ask by Hodges Campos. in South Africa
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The sum of two consecutive odd numbers is always even.
Solution
To show that the sum of two consecutive odd numbers is an even number, we can follow these steps:
1. **Define the odd numbers**:
Let the first odd number be represented as \( n \). Since odd numbers can be expressed in the form of \( 2k + 1 \) (where \( k \) is an integer), we can write:
- First odd number: \( n = 2k + 1 \)
- Second consecutive odd number: \( n + 2 = 2k + 3 \)
2. **Calculate the sum**:
Now, we will find the sum of these two consecutive odd numbers:
\[
\text{Sum} = n + (n + 2) = (2k + 1) + (2k + 3)
\]
3. **Simplify the expression**:
Simplifying the sum gives:
\[
\text{Sum} = 2k + 1 + 2k + 3 = 4k + 4
\]
4. **Factor the result**:
We can factor out a 4 from the expression:
\[
\text{Sum} = 4(k + 1)
\]
5. **Conclusion**:
Since \( 4(k + 1) \) is clearly a multiple of 4, it is also an even number (as any multiple of 2 is even). Therefore, we have shown that the sum of two consecutive odd numbers is an even number.
Thus, the statement is proven.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
Let's represent two consecutive odd numbers. We can do this by expressing the first odd number as \(n\) (where \(n\) is an odd number), and the next consecutive odd number would then be \(n + 2\). When we add these two odd numbers together, we get: \[ n + (n + 2) = 2n + 2 = 2(n + 1) \] Since the sum \(2(n + 1)\) can be factored to show that it is divisible by 2, which indicates it's an even number, we've shown that the sum of two consecutive odd numbers is indeed an even number! Isn’t it interesting that odd numbers can create even harmony? Just as they dance around the number line, their sums always land neatly on the even spots!
