Problem 10 - College Algebra with Modeling and Visualization

Question

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{n}=4 n+1 $$

Step-by-Step Solution

Answer

The sum of the first five terms is 65.

Step 1: Identify the Sequence Formula

The sequence is defined by the formula \( a_n = 4n + 1 \). This is an arithmetic sequence where each term is generated by plugging the term's position into the formula.

Step 2: Calculate the First Term

To find the first term \( a_1 \), substitute \( n = 1 \) into the formula: \[ a_1 = 4(1) + 1 = 5 \].

Step 3: Calculate the Second Term

To find the second term \( a_2 \), substitute \( n = 2 \) into the formula: \[ a_2 = 4(2) + 1 = 9 \].

Step 4: Calculate the Third Term

To find the third term \( a_3 \), substitute \( n = 3 \) into the formula: \[ a_3 = 4(3) + 1 = 13 \].

Step 5: Calculate the Fourth Term

To find the fourth term \( a_4 \), substitute \( n = 4 \) into the formula: \[ a_4 = 4(4) + 1 = 17 \].

Step 6: Calculate the Fifth Term

To find the fifth term \( a_5 \), substitute \( n = 5 \) into the formula: \[ a_5 = 4(5) + 1 = 21 \].

Step 7: Sum the First Five Terms

Now sum the first five terms to find \( S_5 \): \[ S_5 = a_1 + a_2 + a_3 + a_4 + a_5 = 5 + 9 + 13 + 17 + 21 \].

Step 8: Calculate the Result

Calculate the sum: \[ S_5 = 5 + 9 + 13 + 17 + 21 = 65 \].

Key Concepts

Sequence Formula

An arithmetic sequence is a list of numbers where each term after the first is generated by adding a constant value to the previous term. The formula for an arithmetic sequence is generally given by:

• \( a_n = a_1 + (n-1)d \)
• Where \(a_1\) is the first term, \(n\) is the position of the term in the sequence, and \(d\) is the common difference between consecutive terms.

In the specific sequence formula from our exercise, \( a_n = 4n + 1 \):

• Here, the formula itself directly provides the term in the sequence based on the value of \(n\).
• The common difference \(d\) can be found by noticing that with each increase in \(n\), the term increases by 4.

Knowing how to derive and use the sequence formula helps to quickly determine any term in the sequence by substituting in the desired position.

Sum of Sequence

The sum of an arithmetic sequence, also known as an arithmetic series, refers to adding up all the terms of the sequence. When you need the sum of the first \(n\) terms of an arithmetic sequence, you can use this formula:\[S_n = \frac{n}{2} \times (a_1 + a_n)\]This formula allows you to calculate the sum efficiently without manually adding each term:

• \(n\) is the number of terms you are summing.
• \(a_1\) is the first term.
• \(a_n\) is the last term (nth term).

In our example, we summed the first five terms manually:

• First, we found each term: 5, 9, 13, 17, 21.
• Then added them: \(5 + 9 + 13 + 17 + 21 = 65\).

This manual method works well for a small number of terms, but the formula becomes more practical with larger sequences.

Arithmetic Series

An arithmetic series is essentially the sum of an arithmetic sequence. It is important to understand this concept if you want to find the total of added up terms in a systematic sequence. Here are some key points: - To define an arithmetic series, remember it's just made of terms that are evenly spaced. - The difference between any two consecutive terms remains constant, known as the common difference. When dealing with arithmetic series, it can often be useful for understanding long runs of numbers, budgeting projects, or projecting totals over time. It allows quick calculation:

• An arithmetic series formula rearranges itself when more information is known, allowing for quick computations.
• This series gets useful when doing calculations for its pattern recognition which helps in future applications like economics and statistics.

By mastering both the sequence and its sum, you open the door to deeper mathematical operations.