Calculating the Sum of Negative Integers Greater than -100

Understanding the sum of negative integers greater than -100 is a fundamental concept in mathematics that touches on the properties of arithmetic sequences. This topic is essential not only for solving mathematical problems but also for enhancing SEO optimization in online educational content.

Introduction to Arithmetic Series

Before diving into the calculations, it's important to understand the basic concept of an arithmetic series. An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference, denoted by 'd'. In this article, we will solve for the sum of a specific arithmetic series: the negative integers greater than -100.

Identifying the Sequence

The sequence of negative integers greater than -100 starts from -99 and ends at -1. This sequence is represented as:

-99, -98, -97, ..., -2, -1

Here, the first term (a) is -99, and the last term (l) is -1. The common difference (d) is 1, as each term decreases by 1 from the previous term.

Calculation of the Number of Terms

The number of terms (n) in the series can be calculated using the formula:

n l - a 1

Substituting the values:

n -1 - (-99) 1 99 1 100 - 1 99

Therefore, there are 99 terms in the series.

Sum of the Arithmetic Series

The sum (Sn) of an arithmetic series can be calculated using the formula:

S_{n} frac{n}{2} (a l)

Substituting the values we have:

S_{99} frac{99}{2} (-99 (-1))

Simplifying:

S_{99} frac{99}{2} (-100) 99 times (-50) -4950

Thus, the sum of all negative integers greater than -100 is -4950.

Alternative Method

As an alternative, let's consider the sum of the positive integers from 1 to 99:

s sum_{k1}^{99} k frac{99 times 100}{2} 4950

Since the sequence of negative integers from -1 to -99 is the additive inverse of the positive integers from 1 to 99, their sum will be the negative of the sum of the positive integers. Therefore:

-4950

Finally, we reach the same result: the sum of all negative integers greater than -100 is -4950.