Sum Of 3 Consecutive Integers Explained
Ever wondered about the hidden patterns within numbers? Mathematics often reveals elegant simplicities, and one such intriguing concept is the sum of consecutive integers. Understanding how three numbers in a row add up can unlock a deeper appreciation for arithmetic and problem-solving. Let's delve into this fundamental idea, exploring its simple elegance and practical applications. — Whitneywren1 OnlyFans: Uncensored Nude Content
What Are Consecutive Integers?
Before we uncover their sum, let's clarify what "consecutive integers" truly mean. Simply put, these are integers that follow each other in order, with a difference of 1 between each subsequent number.- Examples:
- 1, 2, 3
- 10, 11, 12
- -5, -4, -3
The Simple Formula for Their Sum
When you add three consecutive integers, a fascinating pattern emerges. Let's denote the first integer as 'n'.- The first integer is: n
- The second integer is: n + 1
- The third integer is: n + 2
Combining like terms, we get: 3n + 3
This can be factored to reveal an even simpler form: 3(n + 1)
Notice that '(n + 1)' is always the middle integer! This means the sum of three consecutive integers is always three times the middle integer. — Dallas College: Get Virtual Advising Now!
- Example 1: Sum of 4, 5, 6
- Here, n = 4. The middle integer is 5.
- Using the formula: 3 * (4 + 1) = 3 * 5 = 15.
- Direct sum: 4 + 5 + 6 = 15. It matches!
- Example 2: Sum of 21, 22, 23
- Here, n = 21. The middle integer is 22.
- Using the formula: 3 * (21 + 1) = 3 * 22 = 66.
- Direct sum: 21 + 22 + 23 = 66. Another perfect match!
Why This Formula Works
This elegant property arises from the arithmetic sequence itself. By distributing the "extra" 1 from the second term to the first, and the "extra" 2 from the third term (1 to the first, 1 to the second), you effectively make all three numbers equal to the middle number. For instance, (n), (n+1), (n+2) can be thought of as (n+1 - 1), (n+1), (n+1 + 1). When summed, the -1 and +1 cancel out, leaving 3 times (n+1).Practical Applications and Quick Checks
Understanding this concept isn't just a mathematical curiosity; it's a useful tool in various scenarios:- Problem Solving: If you're given that the sum of three consecutive integers is X, you can quickly find the integers by dividing X by 3 to get the middle integer, then finding the numbers before and after it.
- Divisibility Rule: A number can only be the sum of three consecutive integers if it is perfectly divisible by 3. If a number divided by 3 leaves a remainder, it cannot be formed by adding three consecutive integers.
