You can sum sequential numbers efficiently using a simple formula. This formula, popularized by Carl Gauss, allows for quick calculation of the sum of an arithmetic series.
Understanding the Formula
The formula is: (n / 2)(first number + last number) = sum
Where:
- n is the number of integers in the sequence.
- first number is the starting number of the sequence.
- last number is the ending number of the sequence.
- sum is the total sum of the sequence.
According to a reference dated 01-Apr-2023, this formula is based on the idea that the sum of pairs of consecutive numbers (starting with the first and last numbers) is equal throughout the sequence.
Applying the Formula: Examples
Let's look at some examples to illustrate how to use the formula:
-
Example 1: Sum of numbers from 1 to 10
- n = 10 (There are 10 numbers in the sequence)
- First number = 1
- Last number = 10
- Sum = (10 / 2) (1 + 10) = 5 11 = 55
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Example 2: Sum of even numbers from 2 to 20
- n = 10 (There are 10 even numbers between 2 and 20)
- First number = 2
- Last number = 20
- Sum = (10 / 2) (2 + 20) = 5 22 = 110
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Example 3: Sum of numbers from 5 to 15
- n = 11 (There are 11 numbers in the sequence)
- First number = 5
- Last number = 15
- Sum = (11 / 2) (5 + 15) = 5.5 20 = 110
Practical Insights and Applications
This formula is extremely useful in various scenarios:
- Simplifying calculations: It avoids the need to add each number individually, especially for long sequences.
- Programming: It can be implemented in code for efficient calculation of series sums.
- Problem-solving: Useful in math problems related to arithmetic progressions.
Summary
By using Carl Gauss's formula, (n / 2)(first number + last number) = sum, we can quickly calculate the sum of sequential numbers. This method is based on the observation that the sum of the pairs in consecutive numbers is equal, as highlighted in the provided reference.
