What can we say about the sum of three consecutive integers? Provide evidence to justify your response.
What can we say about the sum of three consecutive integers? Provide evidence to justify your response.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Question:**
What can we say about the sum of three consecutive integers? Provide evidence to justify your response.
**Explanation:**
When discussing three consecutive integers, we can represent them as \( n \), \( n+1 \), and \( n+2 \). The sum of these integers can be expressed as:
\[ n + (n + 1) + (n + 2) = 3n + 3 \]
Simplifying this expression gives:
\[ 3n + 3 = 3(n + 1) \]
This shows that the sum of three consecutive integers is always a multiple of 3.
**Justification with Examples:**
1. Consider the integers 1, 2, and 3:
- Sum = \( 1 + 2 + 3 = 6 \), which is \( 3 \times 2 \).
2. Consider the integers 4, 5, and 6:
- Sum = \( 4 + 5 + 6 = 15 \), which is \( 3 \times 5 \).
3. Consider the integers 7, 8, and 9:
- Sum = \( 7 + 8 + 9 = 24 \), which is \( 3 \times 8 \).
In each case, the sum is divisible by 3, supporting our initial conclusion. Thus, the sum of any three consecutive integers will always result in a multiple of 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27b38c32-736b-4354-8263-82b5dd937d13%2Fa747a3ed-a0ae-4bb8-aed5-b2a584a32e15%2Ffd9pl88_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
What can we say about the sum of three consecutive integers? Provide evidence to justify your response.
**Explanation:**
When discussing three consecutive integers, we can represent them as \( n \), \( n+1 \), and \( n+2 \). The sum of these integers can be expressed as:
\[ n + (n + 1) + (n + 2) = 3n + 3 \]
Simplifying this expression gives:
\[ 3n + 3 = 3(n + 1) \]
This shows that the sum of three consecutive integers is always a multiple of 3.
**Justification with Examples:**
1. Consider the integers 1, 2, and 3:
- Sum = \( 1 + 2 + 3 = 6 \), which is \( 3 \times 2 \).
2. Consider the integers 4, 5, and 6:
- Sum = \( 4 + 5 + 6 = 15 \), which is \( 3 \times 5 \).
3. Consider the integers 7, 8, and 9:
- Sum = \( 7 + 8 + 9 = 24 \), which is \( 3 \times 8 \).
In each case, the sum is divisible by 3, supporting our initial conclusion. Thus, the sum of any three consecutive integers will always result in a multiple of 3.
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