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The sum of a sequence of consecutive numbers.​

In mathematics, calculating the sum of consecutive integers involves adding a series of numbers that follow each other without gaps. These sequences are a subset of arithmetic sequences where the common difference between terms is 1. Understanding how to efficiently compute these sums is fundamental in various mathematical applications.​

Formula for the Sum of Consecutive Integers:

To find the sum of consecutive integers from a starting integer a to an ending integer l, use the formula:​
S = (n/2) × (a + l)​
Where:
S = sum of the consecutive integers
n = number of terms in the sequence, calculated as l - a + 1
a = first term in the sequence
l = last term in the sequence

Example Calculation:

Consider finding the sum of integers from 1 to 100:​
Identify the first term (a) and the last term (l):​
a = 1
l = 100​
Calculate the number of terms (n):​
n = 100 - 1 + 1 = 100​
Apply the formula:​
S = (100/2) × (1 + 100) = 50 × 101 = 5,050​

This method allows for quick computation of sums without the need to add each number individually. Such techniques are widely applicable in algebra and number theory, providing foundational tools for solving more complex mathematical problems.​

For more detailed explanations and examples, refer to resources like BYJU'S article on the sum of consecutive numbers.

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