Problem 38 - A Graphical Approach to College Algebra

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 the First Term and Common Difference

The first term of the sequence, denoted as \(a_1\), is given as 8. The common difference \(d\) is found by subtracting the second term from the first, i.e., \(6 - 8 = -2\).

Step 2: Use the Formula for the Sum of an Arithmetic Sequence

The formula to find the sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is \[ S_n = \frac{n}{2} \cdot (2a_1 + (n-1) \cdot d) \] where \(n\) is the number of terms, \(a_1\) is the first term, and \(d\) is the common difference.

Step 3: Substitute Values into the Formula

Substitute \(n = 10\), \(a_1 = 8\), and \(d = -2\) into the formula: \[ S_{10} = \frac{10}{2} \cdot (2 \cdot 8 + (10 - 1) \cdot (-2)) \] Simplify the expression: \[ S_{10} = 5 \cdot (16 + (-18)) \] \[ S_{10} = 5 \cdot (-2) \]

Step 4: Calculate the Sum

Compute the multiplication: \[ S_{10} = 5 \cdot (-2) = -10 \]Thus, the sum of the first 10 terms is -10.

Key Concepts

Common Difference

The common difference is a crucial element in understanding an arithmetic sequence. In an arithmetic sequence, each term differs from the previous one by a fixed, constant amount, known as the common difference. This difference is the foundation that separates arithmetic sequences from other types of sequences.
If you look at the sequence given in our problem, it starts as 8, 6, 4, and continues similarly. To find the common difference, simply subtract the second term from the first. For this sequence, it would be:

• First term: 8
• Second term: 6
• Common difference (d): 6 - 8 = -2

The common difference can be positive or negative. If it's positive, each subsequent term will increase. If negative, as in our example, each term decreases. Knowing the common difference allows you to quickly find any term in the sequence without listing all the previous ones.

Sum of Arithmetic Sequence

Calculating the sum of an arithmetic sequence involves knowing a few key details about the sequence itself. The general goal is to find the sum of the first few terms, like the first 10 terms in this exercise. The sum of an arithmetic sequence can be found using a specific formula:
- The sum of the first n terms, denoted as \( S_n \), can be calculated using:\[S_n = \frac{n}{2} \cdot (2a_1 + (n-1) \cdot d)\]This formula requires:

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

Using this formula not only simplifies the process of finding the sum but also ensures precision, especially for longer sequences.

Sequence Formula

The sequence formula in arithmetic sequences helps us compute or verify terms without manually adding or listing them. In an arithmetic sequence, the nth term can be found using:\[a_n = a_1 + (n-1) \,d\]Here:

• \( a_n \) is the nth term
• \( a_1 \) is the first term of the sequence
• \( n \) is the term number
• \( d \) is the common difference

This formula allows you to plug in any term number to instantly get its value. As an example, in our sequence, to find the 10th term \( a_{10} \), you use:\[a_{10} = 8 + (10-1)(-2)\]Solving this gives \( a_{10} = -10 \). This precision is particularly useful for verifying terms when following the sequence over multiple steps or when calculating longer sequences.