Sum of Integers from 501 to 600: Simplified Calculation
Calculating the sum of integers from 501 to 600 can seem daunting, but with the right approach, it becomes much simpler. This article aims to provide a clear, step-by-step process using the formula for the sum of an arithmetic series, making it easier for both students and professionals. We will explore different methods to find the sum and highlight the importance of using the appropriate formula.
Understanding the Problem
The problem is to find the sum of all integers between 501 and 600, inclusive. An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. In this case, the common difference is 1.
Method 1: Using the Formula for the Sum of an Arithmetic Series
The formula for the sum of an arithmetic series is given by:
[S_n frac{n}{2}(a l)]
Where:
(S_n) is the sum of the series (n) is the number of terms (a) is the first term (l) is the last termIn this case:
The first term, (a 501) The last term, (l 600) To find the number of terms, we calculate (n 600 - 501 1 100)Substituting these values into the formula:
[S_{100} frac{100}{2} (501 600) 50 times 1101 55050]
Method 2: Using the Sum of an A.P. Formula
The sum of the first (n) terms of an arithmetic progression (A.P.) is given by:
[S_n frac{n}{2} [2a (n - 1)d]]
Where:
(n) is the number of terms (a) is the first term (d) is the common differenceIn this case:
The first term, (a 501) The common difference, (d 1) (n 100)First, we calculate the sum of the first 100 terms (from 1 to 100):
[S_{100} frac{100}{2} [2 times 1 (100 - 1) times 1] 50 times 101 5050]
Adding 500 to each term in the series from 1 to 100 gives us the sum from 501 to 600:
[500 5050 55050]
Method 3: Using the Difference of Two Series
The sum of integers from 501 to 600 can be calculated by finding the sum of integers from 1 to 600 and subtracting the sum of integers from 1 to 500:
Total sum of 1 to 600:
[600 times 601 / 2 180300]
Sum of integers from 1 to 500:
[500 times 501 / 2 125250]
Subtracting the two sums:
[180300 - 125250 55050]
Conclusion
Using the different methods, we have arrived at the same result: the sum of integers from 501 to 600 is 55050.