Problem 28 - Algebra for College Students

Question

Find the sum of all even numbers between 8 and 384 , inclusive. \(\quad 37,044\)

Step-by-Step Solution

Answer

The sum is 37,044.

Step 1: Identify the Sequence of Even Numbers

The first step is to identify the sequence of even numbers starting from 8 up to and including 384. The sequence will be: 8, 10, 12, ..., 384.

Step 2: Determine the Common Difference

Since we are working with even numbers, the common difference between consecutive terms is 2. This means each number is 2 greater than the previous one.

Step 3: Use the Formula for the Sum of an Arithmetic Sequence

The formula to find the sum of an arithmetic sequence is \( S_n = \frac{n}{2} \cdot (a + l) \), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term in the sequence.

Step 4: Calculate the Number of Terms

To find \(n\), substitute the values into the formula \( l = a + (n-1) \cdot d \). Here, \(a = 8\), \(l = 384\), and \(d = 2\). Thus, \(384 = 8 + (n-1) \cdot 2\). Solve for \(n\):\[384 = 8 + 2(n - 1)\]\[384 - 8 = 2(n - 1)\]\[376 = 2(n - 1)\]\[188 = n - 1\]\[n = 189\]

Step 5: Calculate the Sum of the Sequence

Now substituting in \(n = 189\), \(a = 8\), and \(l = 384\) into the sum formula:\[S_n = \frac{189}{2} \cdot (8 + 384)\]\[S_n = 94.5 \cdot 392\]\[S_n = 37,044\]

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence is a simple yet powerful concept in mathematics. It allows us to quickly find the total of a series of numbers that have a consistent difference between them. This is particularly useful when you're dealing with long lists of numbers, like even numbers between 8 and 384.

To calculate the sum, we use the formula:

• \( S_n = \frac{n}{2} \cdot (a + l) \)

Here, \(S_n\) represents the sum of the sequence, \(n\) is the number of terms in the sequence, \(a\) is the first term, and \(l\) is the last term. This formula works because it effectively pairs numbers in the sequence to make calculations easier. For example, when finding the sum from 8 to 384, it simplifies the process by pairing the beginning and end numbers which sum up consistently across the sequence.

In our problem, once you determine that there are 189 terms from 8 to 384, the sum formula elegantly gives us 37,044.

Even Numbers

Even numbers are numbers that can be divided evenly by 2 without any remainder. These include numbers like 2, 4, 6, and so on. In an arithmetic sequence of even numbers, each number increases by the same amount, making them distinctively predictable and easy to work with.

In this exercise, we considered even numbers ranging from 8 to 384. These numbers are a part of an arithmetic sequence, where each even number is obtained by adding 2 to the previous number. This characteristic allows us to use arithmetic sequence formulas to find sums or specific terms within the sequence efficiently.

Understanding that each step or "common difference" in this sequence of even numbers is 2, helps reinforce the arithmetic properties that make even numbers such a foundational part of arithmetic and number theory.

Common Difference

The common difference in an arithmetic sequence is the amount by which consecutive terms increase or decrease. It is a crucial component for understanding and working with arithmetic sequences since it sets the pattern for the entire sequence.

In our specific problem about even numbers, the common difference is 2. This is because every subsequent even number is just 2 units away from its predecessor. Having a common difference not only defines the sequence but also allows us to apply the arithmetic sequence sum formula effectively.

The common difference makes it possible to know the behavior of the sequence. For instance, it's easy to identify the subsequent terms if you know the starting term and the common difference. In this case, identifying a common difference emphasizes the predictability and regularity in arithmetic sequences, making them easy to analyze and compute.