Problem 56 - Beginning and Intermediate Algebra

Question

Write each series using summation notation. 4+8+12+16+20+24+28

Step-by-Step Solution

Answer

The series 4 + 8 + 12 + 16 + 20 + 24 + 28 can be represented using summation notation as: \(\sum_{n=1}^{7} 4n\)

Step 1: Identify the pattern

The given series is: 4, 8, 12, 16, 20, 24, 28 We can see that the numbers are increasing by 4 at each step: 8 - 4 = 4, 12 - 8 = 4, 16 - 12 = 4, and so on.

Step 2: Find the general term of the series

Since this is an arithmetic series with a common difference of 4, we can write its general term as: \(a_n = a_1 + (n - 1)d\), where \(a_n\) is the nth term of the series, \(a_1\) is the first term (which is 4), \(n\) is the index number (which changes from 1 to the number of terms in the series), and \(d\) is the common difference (which is 4). We get: \(a_n = 4 + (n - 1)4\), \(a_n = 4 + 4n - 4\), \(a_n = 4n\).

Step 3: Determine the number of terms in the series

Now, look at the series and count the number of terms: 4, 8, 12, 16, 20, 24, 28 There are a total of 7 terms in the series.

Step 4: Represent the series using summation notation

Using the general term of the arithmetic series and the number of terms, we can represent the series using the summation notation: \(\sum_{n=1}^{7} 4n\) This tells us to add up all the terms of the series starting with n = 1 and ending with n = 7 with the general term 4n. In conclusion, the series 4 + 8 + 12 + 16 + 20 + 24 + 28 can be written using the summation notation as: \(\sum_{n=1}^{7} 4n\)

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers where each term after the first is obtained by adding a fixed amount, known as the common difference, to the previous term. In our given sequence 4, 8, 12, etc., every number increases by 4. This means the series is arithmetic. Simple, right? The pattern in arithmetic series allows us to use a formula easily to find specific terms or sum them up. Identifying that you've got an arithmetic series is the first step in solving problems using summation notation.

General Term

The general term in an arithmetic series is a formula that helps you find any term in the sequence without listing them all out. It's like having a magic wand for series! The formula for the general term is:

• \(a_n = a_1 + (n - 1)d\)

This formula uses:

• \(a_1\), the first term in the series (here that's 4).
• \(d\), the common difference (which we found is 4).
• \(n\), the term position you want to find.

Plug these into the formula to get your general term. In this case, it's simplified to \(a_n = 4n\). No more guesswork!

Common Difference

The common difference is key to understanding an arithmetic series. It's the amount each term increases from the one before. Look closely at 4, 8, 12: subtract one term from the next, like this:

• 8 - 4 = 4
• 12 - 8 = 4

You find that the number you keep getting is 4, which is our common difference. This not only helps spot the pattern in the series but also aids in writing the general term and setting up summation notation. In all arithmetic series, the common difference remains the same between consecutive terms.

Series Representation

Representing a series in summation notation is like condensing a long list into a neat, mathematical expression. For an arithmetic series, it involves using the general term you've derived and showing how many terms there are. In our exercise:

• We have 7 terms: 4, 8, 12, ..., 28.
• We've found that the general term, \(a_n\), is \(4n\).

So, the series starting from the first term to the seventh can be written as:

• \(\sum_{n=1}^{7} 4n\)

This notation tells us to sum all terms from \(n = 1\) to \(n = 7\) for \(4n\). It's clear and concise, capturing the essence of the series without listing each term individually.