Problem 66 - A Graphical Approach to College Algebra

Question

Evaluate each sum. $$\sum_{k=1}^{2000} k$$

Step-by-Step Solution

Answer

The sum is 2001000.

Step 1: Identify the Summation

The exercise requires you to evaluate the sum of the first 2000 positive integers. This can be expressed using the summation formula: \[\sum_{k=1}^{2000} k\]

Step 2: Use the Formula for the Sum of an Arithmetic Series

Recognize that the series \(1 + 2 + 3 + \, \ldots \, + 2000\) is an arithmetic series. The formula to find the sum \(S_n\) of the first \(n\) integers is: \[S_n = \frac{n(n+1)}{2}\] In this situation, \(n = 2000\).

Step 3: Substitute the Values into the Formula

Substitute \(n = 2000\) into the formula for the sum of the first \(n\) integers: \[S_{2000} = \frac{2000(2000 + 1)}{2}\]

Step 4: Calculate the Sum

Now, calculate \(2000 \times 2001\): \[2000 \times 2001 = 4002000\]Divide the result by 2 to obtain the sum: \[\frac{4002000}{2} = 2001000\]

Step 5: Conclusion

The sum of the first 2000 integers is: \[\sum_{k=1}^{2000} k = 2001000\]

Key Concepts

Summation Formula

In mathematics, a summation formula is a way to find the addition of a sequence of numbers. The notation \(\sum_{k=1}^{n} k\) represents the sum of the first \(n\) positive integers. Summation formulas are very useful, especially when dealing with large numbers. Instead of adding each number one by one, you can use a formula to find the answer quickly.
For this exercise, the formula used is for an arithmetic series. But in a more general context, there are many types of summation formulas for different types of sequences.
Whenever you see the sigma \(\Sigma\) symbol in mathematics, it usually indicates summation. The number below the symbol shows where you start summing, and the number above tells you where to end. This concise notation is extremely powerful and reduces complex calculations into simpler math.

Sum of Integers

The sum of the first \(n\) positive integers is a classic formula in arithmetic series. It is given by the equation \(S_n = \frac{n(n+1)}{2}\). This formula provides a quick way to add consecutive numbers.
For example, if you need to find the sum from 1 to 2000, you simply substitute \(n = 2000\) in the formula to get \(S_{2000} = \frac{2000 \times 2001}{2}\). This turns a potentially lengthy addition task into a simple multiplication and division task.
The beauty of this formula is in its simplicity and elegance, which makes it easier to use than adding each number manually. Despite being simple, it's a powerful tool for mathematicians and students.

Mathematical Series

A mathematical series is a sum of terms that follow a specific pattern or rule. In this case, we are dealing with an arithmetic series where each term increases by a constant amount—in this case, 1. The formula for the sum of this series helps illustrate the pattern involved in adding consecutive integers.

Mathematical series can be arithmetic, geometric, or follow other distinctive patterns. Each type has its own formula for finding the sum. Recognizing the type of series you are dealing with can help you pick the right formula. Arithmetic series have a common difference between terms, while geometric series have a common ratio.
The concept of a series extends beyond arithmetic, making it an essential part of advanced math topics like calculus and beyond.