Problem 35 - A Graphical Approach to College Algebra

Question

Find the sum of the first 10 terms of each arithmetic sequence. $$a_{3}=5, a_{4}=8$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 125.

Step 1: Understand the Problem

We are given two terms of an arithmetic sequence: the third term \(a_3 = 5\) and the fourth term \(a_4 = 8\). We need to find the sum of the first 10 terms of this sequence.

Step 2: Determine the Common Difference

Since we have two consecutive terms of the sequence, we can find the common difference \(d\) as follows:\[ d = a_4 - a_3 = 8 - 5 = 3 \]Now we know the common difference is \(d = 3\).

Step 3: Find the First Term of the Sequence

To find the first term \(a_1\), we use the fact that:\[ a_3 = a_1 + 2d \]Substituting the known values, we have:\[ 5 = a_1 + 2(3) \]\[ 5 = a_1 + 6 \]\[ a_1 = 5 - 6 = -1 \]So, the first term \(a_1 = -1\).

Step 4: Use the Sum Formula for the First 10 Terms

The sum of the first \(n\) terms \(S_n\) of an arithmetic sequence is given by:\[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]Here, \(n = 10\), \(a_1 = -1\), and \(d = 3\). Substitute these values:\[ S_{10} = \frac{10}{2} (2(-1) + (10-1) \cdot 3) \]\[ S_{10} = 5 (-2 + 27) \]\[ S_{10} = 5 \times 25 \]\[ S_{10} = 125 \]

Step 5: Conclude the Solution

We have calculated the sum of the first 10 terms of the arithmetic sequence, which is \(S_{10} = 125\).

Key Concepts

Common Difference

The common difference in an arithmetic sequence is a crucial element in understanding the pattern of the sequence. It's the consistent amount by which each term in the sequence increases or decreases from the previous term. For example, in our exercise, we were given that the third term \(a_3 = 5\) and the fourth term \(a_4 = 8\) in the sequence. To find the common difference \(d\), we simply subtract the third term from the fourth term:

• \(d = a_4 - a_3 = 8 - 5 = 3\)

This means that each term in the sequence is 3 more than the preceding term.
Understanding the common difference is important because it allows us to find any term within the sequence and also aids in calculating the sum of the sequence later.

Sum of Arithmetic Sequence

The sum of an arithmetic sequence involves adding up all the terms from the start to a certain term. It's often referred to as the partial sum. The formula to calculate this is straightforward and powerful.Usually, the sum \(S_n\) of the first \(n\) terms is given by:

• \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\)

Where:

• \(n\) is the number of terms
• \(a_1\) is the first term
• \(d\) is the common difference

Using the example, we needed to find the sum of the first 10 terms:

• \(S_{10} = \frac{10}{2} (2(-1) + (10-1) \cdot 3) = 125\)

This illustrates how arithmetic sequences can be easily summed up using the formula, saving time and making calculations efficient.

Arithmetic Sequence Formula

The arithmetic sequence formula helps to find any term in the sequence by knowing the first term and the common difference. The formula for the \(n\)-th term \(a_n\) of an arithmetic sequence is:

• \(a_n = a_1 + (n-1)d\)

Here's why it's useful. Suppose we need to find a term that isn’t given directly, we can just plug our details into this formula.For example, to find the first term \(a_1\) from the given third term \(a_3 = 5\):

• \(a_3 = a_1 + 2d\)
• \(5 = a_1 + 2(3) = a_1 + 6\)
• \(a_1 = 5 - 6 = -1\)

This formula underpins many calculations involving sequences and forms the basis for calculating sums and future terms with ease.