Problem 20 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the arithmetic series. $$ 89+84+79+74+\dots+9+4 $$

Step-by-Step Solution

Answer

The sum of the series is 837.

Step 1: Identify the first and last terms

The first term \(a\) of the series is 89, and the last term \(l\) is 4.

Step 2: Find the common difference

The sequence appears to decrease by a constant amount, which is the common difference \(d\). Calculate this by subtracting the second term from the first term: \(84 - 89 = -5\). So, the common difference \(d = -5\).

Step 3: Determine the number of terms

To find the number of terms \(n\), use the formula for the n-th term of an arithmetic sequence: \(l = a + (n - 1) \cdot d\). Plug in the known values: \[4 = 89 + (n - 1) \cdot (-5)\]Solving for \(n\):\[4 = 89 - 5n + 5\]\[4 = 94 - 5n\]\[5n = 90\]\[n = 18\]So, there are 18 terms in the series.

Step 4: Use the formula to find the sum of the series

The formula for the sum \(S_n\) of an arithmetic series is given by \[S_n = \frac{n}{2} \times (a + l)\]Substitute the values \(n = 18\), \(a = 89\), and \(l = 4\) into the formula:\[S_{18} = \frac{18}{2} \times (89 + 4)\]\[S_{18} = 9 \times 93\]\[S_{18} = 837\]

Key Concepts

Sum of series

When you want to find the sum of an arithmetic series, you're dealing with adding up all the terms in a sequence. For the given problem, the terms reduce consistently by a set number, which simplifies the process of summation. The formula to calculate the sum of an arithmetic series is:

• \( S_n = \frac{n}{2} \times (a + l) \)

Here, \( n \) represents the number of terms, \( a \) is the first term, and \( l \) is the last term of the series. By using this formula, you can quickly find the total of all terms.
In our example, with 18 terms, the first term being 89 and the last one being 4, we apply the formula as follows:

• \( S_{18} = \frac{18}{2} \times (89 + 4) \)
• \( S_{18} = 9 \times 93 \)
• \( S_{18} = 837 \)

So, the sum of this series is 837. This method is helpful to quickly sum up large sequences without manually adding every term.

Common difference

The common difference is a key characteristic of an arithmetic sequence. It tells you how much each term in the series decreases or increases by. In an arithmetic sequence, this value stays constant between consecutive terms.
To find the common difference \( d \), you simply subtract any term from the term that follows it. In our example problem, you calculate it like this:

• \( d = 84 - 89 \)
• \( d = -5 \)

This means each term is 5 less than the previous term. Understanding the common difference is crucial because it allows you to write the formula for any term in the series:

• \( a_n = a + (n-1) \times d \)

Here, \( a_n \) is the n-th term, \( a \) is the first term, and \( d \) is our common difference. Recognizing this pattern opens up the path to more complex calculations involving series.

Number of terms

Calculating the number of terms in an arithmetic series is vital to finding its sum or any n-th term. The formula to find the number of terms \( n \) is derived from the formula that defines any term in the series:

• \( l = a + (n-1) \times d \)

In this equation, \( l \) is the last term, \( a \) is the first term, and \( d \) is the common difference. To find \( n \), you rearrange this formula. Let’s solve it using our example:

• Start with \( 4 = 89 + (n-1) \times (-5) \)
• Then, rearrange to find \( n \): \( 4 = 94 - 5n \)
• Solve for \( n \): \( 5n = 90 \rightarrow n = 18 \)

This demonstrates that there are 18 terms in the series. Knowing the exact count of terms is essential for any further calculations, such as determining the total sum of the sequence.