Problem 13 - Algebra for College Students

Question

Find the sum of the first 150 positive even whole numbers. \(\quad 22,650\)

Step-by-Step Solution

Answer

The sum of the first 150 positive even numbers is 22,650.

Step 1: Understand the Sequence

The sequence of positive even whole numbers starts with 2, 4, 6, ..., and continues increasing by 2 each time. We need to find the sum of the first 150 numbers in this sequence.

Step 2: Identify the Sequence Formula

Recognize that the n-th term of the sequence of even numbers can be written as \( a_n = 2n \) where \( n \) is the position of the term in the sequence. For 150 terms, the last term \( a_{150} = 2 \times 150 = 300 \).

Step 3: Use Sum Formula for Arithmetic Series

The sum of an arithmetic series is given by the formula \( S_n = \frac{n}{2} \times (a_1 + a_n) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. Here, \( n = 150 \), \( a_1 = 2 \), and \( a_{150} = 300 \).

Step 4: Substitute and Calculate

Substitute the known values into the formula:\[S_{150} = \frac{150}{2} \times (2 + 300)\]\[S_{150} = 75 \times 302\]Perform the multiplication to find the sum.

Step 5: Finalize and Verify the Sum

Calculate \( 75 \times 302 \) which equals 22,650.Verify by checking each step logically aligns with the derived sequence patterns and arithmetic properties. This confirms that the sum of the first 150 positive even whole numbers is indeed 22,650.

Key Concepts

Understanding Even Numbers

Even numbers are integers that can be divided by 2 without leaving a remainder. These numbers are critically important in math as they follow simple patterns and properties. The sequence of even numbers starts with 2 and continues in an obvious pattern: 2, 4, 6, 8, and so on.
Understanding even numbers helps solve a variety of problems, such as calculating sums or determining sequences. Here's a quick tip:

• To find any even number in the sequence, simply multiply 2 by the position of the number. For example, the 5th even number is 2 multiplied by 5, which is 10.

This simplicity can be incredibly helpful, especially when you need to work with longer sequences or perform calculations.

Fundamentals of Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant to the previous term. This constant is called the common difference.
If we look at even numbers, you'll notice they form an arithmetic sequence where the common difference is always 2. For instance:

• The sequence 2, 4, 6, 8 has a common difference of 2 because each number is larger than the previous one by 2.

Knowing this helps you predict and calculate the terms of the sequence quickly, which is precisely what we need when finding sums or specific terms in a lengthy series.

Sum Formula for Arithmetic Series

Calculating the sum of an arithmetic series involves using the sum formula. The formula is particularly useful when you want to find the sum of a large number of terms without individually adding each one. The formula is written as:

• \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]

Here's how it works in practice:

• \(n\) is the number of terms in the sequence.
• \(a_1\) is the first term, and \(a_n\) is the last term.

In the context of summing the first 150 even numbers, these apply as:

• \(n = 150\)
• \(a_1 = 2\)
• \(a_{150} = 300\)

Substituting these into the formula gives us:

• \[ S_{150} = \frac{150}{2} \times (2 + 300) \]

This method is not only efficient but also ensures accuracy in your results, as shown by the final calculation of 22,650 for the sum of the first 150 even numbers.