Problem 37 - A Graphical Approach to College Algebra

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$5,9,13, \dots$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 230.

Step 1: Identify the Common Difference

An arithmetic sequence has a common difference which is constant between consecutive terms. Here, the difference between the first two terms is \(9 - 5 = 4\). Therefore, the common difference \(d = 4\).

Step 2: Identify the First Term

The first term of the sequence, denoted as \(a_1\), is the starting point of the sequence. Here, the first term \(a_1 = 5\).

Step 3: Calculate the 10th Term

To find the 10th term \(a_{10}\) of an arithmetic sequence, use the formula \(a_n = a_1 + (n-1)d\). Substitute \(n = 10\), \(a_1 = 5\), and \(d = 4\) into the formula:\[a_{10} = 5 + (10-1) imes 4 = 5 + 36 = 41\]

Step 4: Use the Arithmetic Sum Formula

The sum of the first \(n\) terms \(S_n\) of an arithmetic sequence can be calculated with the formula \(S_n = \frac{n}{2} (a_1 + a_n)\). Here, \(n = 10, a_1 = 5, \) and \(a_{10} = 41\). Substitute these values:\[S_{10} = \frac{10}{2} (5 + 41) = 5 imes 46 = 230\]

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial element that defines the pattern between consecutive terms. It is simply the difference between any two successive terms in the sequence.
For example, consider the sequence 5, 9, 13, … Here, the difference between the first and second term is calculated as follows:

• Subtract the first term from the second: \(9 - 5 = 4\).

This constant value, 4 in this case, is known as the common difference, often denoted by the symbol \(d\). Knowing this value helps in predicting the next terms of the sequence and calculating other required elements, like the nth term or the sum of the sequence.

First Term

The first term of an arithmetic sequence, denoted as \(a_1\), is the term where the sequence begins. This term is foundational because it sets the starting point for the entire sequence.
In our sequence example of 5, 9, 13, … , the first term \(a_1\) is easily identified as 5. This initial value is vital for any further calculations, such as finding the nth term or the sum of terms in the sequence.

• For the given sequence, the first term \(a_1 = 5\).

Remembering the first term is important because it directly influences how the sequence develops.

Arithmetic Sum Formula

The arithmetic sum formula allows us to find the sum of the first few terms of an arithmetic sequence efficiently. The sum of the first \(n\) terms, represented as \(S_n\), can be calculated using this formula:

• \(S_n = \frac{n}{2} (a_1 + a_n)\)

Here, \(n\) is the number of terms to be added, \(a_1\) is the first term, and \(a_n\) is the nth term.
For our sequence 5, 9, 13, ..., we want to find the sum of the first 10 terms.

• Substitute \(n = 10\), \(a_1 = 5\), and \(a_{10} = 41\):\[S_{10} = \frac{10}{2} (5 + 41) = 5 \times 46 = 230\]

This result, 230, is the sum of the first 10 terms of the sequence. Utilizing this formula simplifies what could otherwise be a tedious process of adding each term individually.