Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$8,6,4, \dots$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is -10.

Step 1: Identify Parameters

The given arithmetic sequence is 8, 6, 4,...The first term \(a_1 = 8\).The common difference \(d = 6 - 8 = -2\).The number of terms \(n = 10\).

Step 2: Use Sum Formula

Use the formula for the sum of the first \(n\) terms in an arithmetic sequence:\[S_n = \frac{n}{2} (2a_1 + (n-1)d)\].

Step 3: Insert Known Values

Substitute the known values \(a_1 = 8\), \(d = -2\), and \(n = 10\) into the formula: \[S_{10} = \frac{10}{2} (2 \times 8 + (10-1)(-2))\].

Step 4: Solve Inside the Parentheses

Calculate the expression inside the parentheses:\[2 \times 8 + (10-1)(-2) = 16 + 9 imes (-2)\].Simplifying further, we get:\[16 + (-18) = -2\].

Step 5: Compute Sum

Substitute \(-2\) back into the sum formula:\[S_{10} = \frac{10}{2} \times (-2)\].Calculate the sum:\[S_{10} = 5 \times (-2) = -10\].

Key Concepts

Common Difference in Arithmetic Sequences

In an arithmetic sequence, the common difference is a fundamental aspect that defines the pattern of the sequence. It is the difference between any two consecutive terms of the sequence. This difference is constant throughout the entire sequence, which is why it is called "common."

• To find the common difference, subtract the first term from the second term.
• You can also subtract any term from the next one in the sequence to confirm consistency.
• In our example, the sequence is 8, 6, 4, ... and the common difference, denoted as \(d\), is \(6 - 8 = -2\).

The negative common difference indicates that the sequence decreases as you move from one term to the next. It is crucial to identify this common difference correctly, as it plays a significant role in determining other properties of the sequence, such as the sum of a certain number of terms.

Sum Formula for Arithmetic Sequence

When calculating the total sum of terms in an arithmetic sequence, the Sum Formula is the tool to use. This formula allows you to find the sum of the first \(n\) terms of the sequence quickly and efficiently. The formula is given by:
\[S_n = \frac{n}{2} (2a_1 + (n-1)d)\]
Here’s how to apply it:

• \(S_n\) represents the sum of the first \(n\) terms.
• \(n\) is the number of terms you want to sum up.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.

Let's consider the example sequence: 8, 6, 4, ... with a common difference \(d = -2\) and the first term \(a_1 = 8\).
For 10 terms, inserting these values into the formula:\(S_{10} = \frac{10}{2} (2 \times 8 + (10-1)(-2))\).
Solving this step-by-step helps to avoid errors and leads to finding the correct total sum.

Understanding Arithmetic Series

An arithmetic series is simply the sum of the terms in an arithmetic sequence. Each sequence has its respective series, which has crucial applications in various fields such as finance, mathematics, and physics.

• The series helps in finding the total sum of all elements from the first term to the nth term of a sequence.
• Knowing how to compute an arithmetic series can aid in budgeting, calculating interest over time, or even deducing historical data trends.

Using the sum formula simplifies this process by providing a straightforward way to compute the series without having to add each term manually.
In our example:

• The arithmetic sequence is 8, 6, 4, ..., which forms an arithmetic series when summed: \(8 + 6 + 4 + ... + nth \, term\).
• The calculated sum for the first 10 terms turned out to be \(-10\).

This result demonstrates how the understanding of arithmetic series and its sum formula leads to accurate and efficient calculations.