Problem 54 - Algebra 2

Question

Find the sum of each series. $$ \sum_{n=1}^{3}(5 n-2) $$

Step-by-Step Solution

Answer

The sum of the series is 24.

Step 1: Understand the series

The series given is \( \sum_{n=1}^{3}(5n-2) \). This means we need to evaluate and add up the terms of the series for \( n = 1, 2, \) and \( 3 \).

Step 2: Calculate individual terms

We calculate each term separately by substituting \( n \) into the expression \( 5n - 2 \). For \( n = 1 \), the term is \( 5 \times 1 - 2 = 3 \). For \( n = 2 \), the term is \( 5 \times 2 - 2 = 8 \). For \( n = 3 \), the term is \( 5 \times 3 - 2 = 13 \).

Step 3: Sum the terms

Now, add the individual terms obtained from Step 2: \( 3 + 8 + 13 = 24 \). This is the sum of the series.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is found by adding a constant, called the common difference, to the previous term. In simpler terms, it's like counting by a specific number from a starting point. For example, if you start at 3 and keep adding 5, you get 3, 8, 13, 18, and so on. The uniform addition of 5 is the key feature of an arithmetic series.

Understanding an arithmetic series is all about recognizing this pattern. To find the sum of an arithmetic series, you can use the formula:\[S_n = \frac{n}{2} \times (a_1 + a_n)\]where:

• \( S_n \) is the sum of the series,
• \( n \) is the number of terms,
• \( a_1 \) is the first term,
• \( a_n \) is the last term.

In the exercise, there is an arithmetic progression, starting from 3, with a regular addition of 5 until reaching three terms. By calculating each term and then summing them, we arrive at the solution.

Summation Notation

Summation notation is a concise and efficient way to represent the sum of a series. It uses the Greek letter sigma (\( \Sigma \)) to denote the sum. Often, you'll see something like \( \sum_{n=1}^{k} a_n \), which tells you to sum all terms from \( n = 1 \) to \( n = k \).

The formula in our example, \( \sum_{n=1}^{3}(5n-2) \), instructs us to replace \( n \) with 1, 2, and 3 respectively. For each value of \( n \), calculate the result of \( 5n-2 \), and then add these results together. This is essentially how summation notation streamlines the process of writing and performing sum calculations in mathematics.

Summation notation is extremely helpful in dealing with expressions where the same operation needs to be performed repeatedly for different values. By giving a clear rule about the number of terms and how to calculate them, it makes many algebraic processes easier to understand and manage.

Algebraic Expressions

Algebraic expressions are mathematical phrases that use numbers, variables, and operational symbols. They help express mathematical relationships in a generalized form. In the given exercise, the expression \( 5n-2 \) is an algebraic expression, where "n" is a variable.

Let's break this down:

• The number 5, known as the coefficient, is multiplied by the variable \( n \).
• From this product, 2 is subtracted to get each term of the series.

Being comfortable with expressions like \( 5n-2 \) is fundamental to working with series. By substituting different values of \( n \), you determine the specific values for each term in the series.

Understanding algebraic expressions is crucial because they form the basis for creating equations that model real-world scenarios. They provide a bridge between abstract math and its practical applications, allowing us to calculate and solve problems more flexibly.