Problem 9 - College Algebra with Modeling and Visualization

Question

For the given \(a_{n},\) calculate \(S_{5}.\) $$ a_{n}=2 n-1 $$

Step-by-Step Solution

Answer

The sum of the first 5 terms, \( S_5 \), is 25.

Step 1: Understand the Sequence Formula

Recognize that the given formula for the sequence is \( a_n = 2n - 1 \). This formula will generate the terms of the sequence.

Step 2: Find the First 5 Terms of the Sequence

Use the formula \( a_n = 2n - 1 \) to calculate the first five terms: \( a_1 = 2(1) - 1 = 1 \), \( a_2 = 2(2) - 1 = 3 \), \( a_3 = 2(3) - 1 = 5 \), \( a_4 = 2(4) - 1 = 7 \), and \( a_5 = 2(5) - 1 = 9 \).

Step 3: Calculate the Sum of the First 5 Terms

Sum the first five terms: \( S_5 = a_1 + a_2 + a_3 + a_4 + a_5 \). Place the values into the equation: \( S_5 = 1 + 3 + 5 + 7 + 9 \).

Step 4: Summation of the Calculated Terms

Add the sequence: \( 1 + 3 = 4 \), \( 4 + 5 = 9 \), \( 9 + 7 = 16 \), and \( 16 + 9 = 25 \). Therefore, \( S_5 = 25 \).

Key Concepts

Sum of a Sequence

When working with arithmetic sequences, one key aspect you might encounter is finding the sum of a certain number of terms. The sum of a sequence, like the problem where we calculated the sum \( S_5 \), is an essential concept in arithmetic sequences and mathematics in general. In this example, we were tasked to sum the first five terms of the sequence defined by the formula \( a_n = 2n - 1 \).

It is important to understand that each term in the sequence is derived from the sequence formula. For the first five terms, we calculated \( a_1 = 1, a_2 = 3, a_3 = 5, a_4 = 7, \) and \( a_5 = 9 \). Once you've calculated these values, you simply add them together:

• Step 1: Add the first two terms \( a_1 + a_2 \): \( 1 + 3 = 4 \)
• Step 2: Continue adding: \( 4 + 5 = 9 \)
• Step 3: Add the next term: \( 9 + 7 = 16 \)
• Step 4: Finally add the last term: \( 16 + 9 = 25 \)

In arithmetic sequences like these, you can find formulas specifically for calculating n-term sums, which are very useful in saving time, especially if you're working with large sequences.

Sequence Formula

A sequence formula is a mathematical way to describe a pattern in a sequence of numbers. For an arithmetic sequence, the formula usually takes the form \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference between terms. However, in this particular problem, the sequence formula given is \( a_n = 2n - 1 \).

This formula generates each term of the sequence by plugging in the term number \( n \). Let's see how it works step-by-step:

• For \( n = 1 \), the formula gives \( a_1 = 2(1) - 1 = 1 \)
• For \( n = 2 \), \( a_2 = 2(2) - 1 = 3 \)
• For \( n = 3 \), \( a_3 = 2(3) - 1 = 5 \)
• For \( n = 4 \), \( a_4 = 2(4) - 1 = 7 \)
• For \( n = 5 \), \( a_5 = 2(5) - 1 = 9 \)

As you can see, this sequence formula is designed to generate odd numbers starting from 1. Every arithmetic sequence has such a formula, which makes it easy to find specific terms without needing to add repeatedly. Understanding and identifying the sequence formula for any arithmetic sequence is crucial for analyzing and solving related problems effectively.

Mathematical Series

In mathematics, a series is the sum of the terms of a sequence. Essentially, when you add all the terms in a sequence, you get a mathematical series. This is an important concept, especially when dealing with arithmetic or geometric progressions. In the provided exercise, when we add the sequence terms \( 1, 3, 5, 7, \) and \( 9 \), we are creating a series.

A key aspect of solving series problems involves recognizing the type of sequence (arithmetic or geometric) and applying the appropriate methods or formulas to find a solution. Let's look more closely at how an arithmetic series functions:

• The formula for the sum of the first \( n \) terms of an arithmetic sequence is \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.
• In our example, with the first term \( a_1 = 1 \) and the fifth term \( a_5 = 9 \), the sequence is also an arithmetic sequence with a common difference.
• We calculated the manual sum but note that such formulas allow quicker calculations for larger quantities of terms.

Understanding series, particularly the processes and formulas used in calculations, is very valuable for efficient problem-solving across various mathematical areas.