Problem 40 - College Algebra

Question

Find \(2+4+6+8+\dots+200,\) the sum of the first 100 positive even integers.

Step-by-Step Solution

Answer

The sum of the first 100 positive even integers is 10100.

Step 1: Identify the values to be used in the formula

The first term \( a_1 \) is 2, the last term \( a_n \) is 200 and the sequence has 100 terms \( n = 100 \).

Step 2: Use the arithmetic series sum formula

Substitute \( n, a_1, a_n \) into the sum of an arithmetic series formula, \( S_n = \frac{n}{2} (a_1 + a_n) \)

Step 3: Calculate the sum

Substitute \( n = 100, a_1 = 2, a_n = 200 \) into the formula and calculate the sum: \( S_n = \frac{100}{2} (2 + 200) = 50 * 202 = 10100 \)

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members is a constant called the common difference, denoted as 'd'. In the simplest terms, it means in an AP you will add (or subtract) the same value again and again starting from the first term to get the rest of the terms.

For example, in the given exercise, to find the sum of the first 100 positive even integers, the sequence starts with 2 and the common difference is 2, as each term is an increment of 2 on the previous one (2, 4, 6, 8, ...). The AP is described as a linear pattern, which makes calculations systematic and predictable, especially useful when working with a large number of terms.

Series and Sequences

While both terms are often used interchangeably, there's a slight difference between a sequence and a series in mathematics. A sequence refers to an ordered list of numbers that are typically defined by some function or rule. For instance, an arithmetic sequence is generated by adding the common difference to the previous term.

A series, on the other hand, is what you get when you add up the terms of a sequence. It's the sum of a sequence, which in our exercise is the addition of all the even numbers from 2 to 200. Understanding the distinction is essential when dealing with complex mathematical problems, as it helps in organizing the approach for solving them. In our exercise, we are concerned with finding the sum of a series created by an arithmetic sequence.

Sum of Even Integers

When dealing with even integers, we are looking at a very particular type of arithmetic sequence where the common difference is 2. Calculating the sum of even integers follows the same principle as any arithmetic series but requires recognizing that only every second integer will be included in the series.

The formula for the sum of an arithmetic series is also applicable here and greatly simplifies the calculation. By knowing the first term, the number of terms, and the last term, the sum can be quickly found. In the provided exercise, the sum \( S_n = \frac{n}{2} (a_1 + a_n) \) helps in computing the total of the first 100 positive even integers without needing to add each individual term. It's a convenient method that saves time and reduces the potential for errors in long calculations.