Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{1}=-8, a_{10}=-1.25$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is -46.25.

Step 1: Identify the First Term and the Tenth Term

Given in the problem, the first term of the arithmetic sequence, \(a_1\), is \(-8\) and the tenth term, \(a_{10}\), is \(-1.25\).

Step 2: Determine the Common Difference

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(d\) is the common difference. Plug in the values for the tenth term: \(-1.25 = -8 + 9d\). Solve for \(d\): \[9d = -1.25 + 8\] \[9d = 6.75\] \[d = \frac{6.75}{9} = 0.75\].

Step 3: Apply the Sum Formula for the Arithmetic Series

The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n = \frac{n}{2} (a_1 + a_n)\). For this sequence, \(n = 10, a_1 = -8, a_{10} = -1.25\). Substitute these into the sum formula: \[S_{10} = \frac{10}{2} (-8 - 1.25)\] \[S_{10} = 5 \times (-9.25)\] \[S_{10} = -46.25\].

Key Concepts

Common Difference

In an arithmetic sequence, the difference between consecutive terms remains constant, and this is known as the **common difference**. It is a crucial element as it determines how the sequence progresses. We use the common difference symbolized by the letter \(d\) in mathematical formulas.
To find the common difference if you know two terms in the sequence, use the formula for the \(n\)-th term:

• General Formula: \(a_n = a_1 + (n-1)d\)
• Common Difference Formula: \(d = \frac{a_n - a_1}{n-1}\)

By solving for \(d\), you can calculate how much each term increases. In our example, we calculated \(d\) by knowing \(a_1\) and \(a_{10}\), hence \(d = \frac{6.75}{9} = 0.75\). This value indicates that each term in the sequence is 0.75 greater than the previous one.

Sum Formula

When you want to find the **sum of a certain number of terms** in an arithmetic sequence, the sum formula will be your best friend. This formula provides a convenient way to sum up the terms without having to add each one individually.
The sum of the first \(n\) terms, denoted as \(S_n\), of an arithmetic sequence can be calculated using:

• Sum Formula: \(S_n = \frac{n}{2} (a_1 + a_n)\)

In this formula, \(n\) is the number of terms to sum, \(a_1\) is the first term, and \(a_n\) is the \(n\)-th term. Applying this to our exercise, we calculated \(S_{10}\) with known values for \(n=10\), \(a_1=-8\), and \(a_{10}=-1.25\), giving us \(S_{10} = 5 \times (-9.25) = -46.25\). This tells us that the sum of the first 10 terms is -46.25.

Series

A **series** refers to the sum of the terms of a sequence. In arithmetic sequences, the series provides a way to understand the overall effect of adding successive terms.
For arithmetic sequences, calculating the series involves using the sum formula, which efficiently provides the total sum of all terms considered in the sequence. Let's review key points of an arithmetic series:

• The sequence listed as a sum is what we call a series.
• The arithmetic series formula helps manage large series quickly.

The series in our given problem involved adding ten successive terms starting at -8 and ending at -1.25. By finding the sum as -46.25, the series completion showed how these terms together influenced the total result.