Problem 53 - A Graphical Approach to College Algebra

Question

Use a formula to find the sum of each arithmetic series. $$1+2+3+4+\dots+50$$

Step-by-Step Solution

Answer

The sum of the series is 1275.

Step 1: Identify the components of the series

Recognize that the given series is an arithmetic series where the first term \(a_1 = 1\) and the last term \(a_n = 50\). The total number of terms \(n\) can be found by noticing that the series goes from 1 to 50, hence \(n = 50\).

Step 2: Use the formula for the sum of an arithmetic series

The formula for the sum \(S_n\) of an arithmetic series is: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]Substitute the known values into the formula: \[ S_{50} = \frac{50}{2} \times (1 + 50) \]

Step 3: Calculate the sum

Perform the calculations in the formula. First, simplify inside the parentheses:\[ 1 + 50 = 51 \]Then, calculate the sum:\[ S_{50} = 25 \times 51 \]Finally, compute the product:\[ S_{50} = 1275 \]

Key Concepts

Sum of Arithmetic Series

An arithmetic series is simply the sum of an arithmetic sequence. When you look at a sequence where each term increases by a constant amount, you have an arithmetic sequence. For example, in the sequence 1, 2, 3, 4,... up to 50, each number increases by 1.
The sum, or total, of such a sequence is called an arithmetic series. To find this sum, you can use a special formula that quickly gives you the total without adding every single number.
Understanding how to sum these numbers efficiently is a powerful tool, whether you're calculating a simple sequence or encountering more complex mathematical scenarios in the future. It saves time and reduces chances for errors that might come from manually adding each term.

Arithmetic Series Formula

The formula to find the sum of an arithmetic series is helpful because it provides a quick way to calculate the total. The formula is structured as:

• \( S_n = \frac{n}{2} \times (a_1 + a_n) \)
• • \( S_n \) is the sum of the series
  • \( n \) is the number of terms
  • \( a_1 \) is the first term
  • \( a_n \) is the last term

By substituting your known values into this formula, you can find the total sum without hassle. In our exercise, you use the starting term (1), the ending term (50), and the number of terms (50) to plug into the formula. This leads to:

• \( S_{50} = \frac{50}{2} \times (1 + 50) \)
• Calculate to get \( S_{50} = 1275 \)

This method simplifies complex series into straightforward calculations, illustrating the elegance of arithmetic principles.

Mathematics Education

In learning mathematics, particularly topics like arithmetic series, it's essential to build a strong foundational understanding. Many students find joy in discovering that mathematics often has elegant solutions. These solutions are accessible through understanding and applying the right formulas and concepts.
Education in mathematics doesn't just teach you how to solve equations but also how to think logically and critically. When working through an arithmetic series, for example, students learn how to:

• Identify patterns and sequences
• Apply mathematical formulas and principles
• Develop problem-solving skills

As you continue to explore more about arithmetic series and beyond, you'll gain confidence in both your mathematical abilities and your overall analytical skills. This is the true goal of mathematics education: to empower you with the tools needed for both academic success and everyday problem-solving.