Problem 58 - Beginning and Intermediate Algebra

Question

Find \(S_{8}\) for each arithmetic sequence described below. $$a_{n}=4 n+1$$

Step-by-Step Solution

Answer

The sum of the first 8 terms of the arithmetic sequence \(a_{n}=4n+1\) is \(S_{8}=152\).

Step 1: Find the first 8 terms of the sequence

To find the first 8 terms, plug in the values of \(n\) from 1 to 8 into the formula \(a_{n}=4n+1\). \(a_{1}=4(1)+1=5\) \(a_{2}=4(2)+1=9\) \(a_{3}=4(3)+1=13\) \(a_{4}=4(4)+1=17\) \(a_{5}=4(5)+1=21\) \(a_{6}=4(6)+1=25\) \(a_{7}=4(7)+1=29\) \(a_{8}=4(8)+1=33\)

Step 2: Identify the number of terms, the first term and the last term

The number of terms in the sequence (\(n\)) is 8. The first term of the sequence (\(a_{1}\)) is 5 and the last term (\(a_{8}\)) is 33.

Step 3: Calculate the sum of the first 8 terms using the arithmetic series formula

The formula for the sum of an arithmetic series is given by: $$S_{n}=\frac{n}{2} (a_{1}+a_{n})$$ In this case, \(n=8\), \(a_{1}=5\), and \(a_{8}=33\). Plugging these values into the formula, we get: $$S_{8}=\frac{8}{2} (5+33)$$

Step 4: Evaluate the expression to find the sum of the 8 terms

Simplify the expression to find the sum: $$S_{8} = 4 (38)$$ $$S_{8} = 152$$ Therefore, the sum of the first 8 terms of the arithmetic sequence is 152.

Key Concepts

Sum of Arithmetic Series

The sum of an arithmetic series is a useful concept when dealing with sequences where you need to add together multiple terms. To calculate the sum of an arithmetic series, there's a specific formula that comes in handy. This formula is as follows:
\[S_n = \frac{n}{2} (a_1 + a_n)\]where:

• \(S_n\) represents the sum of the first \(n\) terms in the series.
• \(n\) is the total number of terms.
• \(a_1\) is the first term.
• \(a_n\) is the nth term or the last term you are considering in your sum.

By using this formula, you can easily calculate the sum of terms without having to add each one individually. It's particularly handy in longer sequences.Understanding this formula is crucial, as it allows you to efficiently find the sum in arithmetic progressions.

Sequence Formula

In arithmetic sequences, each term increases or decreases by a constant value called the common difference. The sequence formula used to determine the \(n\)th term of an arithmetic sequence is:
\[a_n = a_1 + (n-1) \, d\]where:

• \(a_n\) is the nth term.
• \(a_1\) is the first term.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference.

Using this formula, one can find any term in the sequence if they know the first term and the common difference. For example, if the formula for a sequence is \(a_n = 4n + 1\), you simply plug in values of \(n\) to find specific terms. This straightforward formula helps in identifying and predicting terms in a sequence.

Arithmetic Progression

Arithmetic progression is a sequence of numbers where each term after the first is determined by adding a constant called the common difference to the previous term.
An arithmetic progression has a very structured approach:

• The difference between consecutive terms (\(d\)) is constant.
• It is characterized by a linear pattern, easy to recognize and compute.

Such sequences can be efficiently analyzed using formulas, where the first term (\(a_1\)) and the common difference (\(d\)) play key roles. This consistent change makes arithmetic progressions simple to work with and perfect for applying summation formulas or sequence equations. They are widely used in various mathematical and real-world applications, providing both simplicity and predictability.