Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

The first two terms of the arithmetic sequence are given. Find the missing term. Use the table feature of a graphing utility to verify your results. $$a_{1}=5, \quad a_{2}=11, \quad a_{10}=\square$$

Step-by-Step Solution

Answer

The tenth term of the given arithmetic sequence is 59.

Step 1: Identify Pattern and Calculate Common Difference

In an arithmetic sequence, every term differs from the previous one by a constant difference, known as the common difference. This is calculated as \(d = a_{2} - a_{1}\), where \(a_{2}\) is 11 and \(a_{1}\) is 5. Hence, \(d = 11 - 5 = 6\). Therefore, the common difference of this arithmetic sequence is 6.

Step 2: Apply Arithmetic Sequence Formula

The formula to find the \(n^{th}\) term in an arithmetic sequence is given by \(a_{n} = a_{1} + (n-1) * d\). Here, we're looking for the \ \(a_{10}\), so \(n = 10\), \(a_{1} = 5\), and \(d = 6\). Substitute these values into the formula, to find \(a_{10}\).

Step 3: Calculate the tenth term

Substituting the known values into the arithmetic sequence formula, we obtain \(a_{10} = 5 + (10-1) * 6 = 5 + 54 = 59\). So, the tenth term of this sequence is 59.

Key Concepts

Common Difference

In an arithmetic sequence, each term differs from the previous one by a constant value. This value is known as the **common difference**. It's a key property that defines the behavior of the sequence. For instance, in the given sequence, the first term is 5 and the second is 11. The common difference can be calculated using the formula:

• \(d = a_2 - a_1\)

Plugging our values into the formula, we get:

• \(d = 11 - 5 = 6\)

So, the common difference for this arithmetic sequence is 6. This means each term increased by 6 to get the next term. Understanding this helps in predicting future terms.

Nth Term Formula

The **nth term formula** is used to find any term in an arithmetic sequence. The formula is written as:

• \(a_n = a_1 + (n-1) \times d\)

Where:

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

For the tenth term \(a_{10}\), we substitute the known values (\(a_1 = 5\), \(d = 6\), and \(n = 10\)) into the formula:

• \(a_{10} = 5 + (10-1) \times 6\)
• \(a_{10} = 5 + 54 = 59\)

Hence, the tenth term of the sequence is 59. This formula is fundamental for quickly finding any term in an arithmetic sequence without listing all terms.

Graphing Utility

A **graphing utility** is a tool that can be used to visually verify solutions of arithmetic sequences and other mathematical problems. With it, you can create tables and graphs to observe the behavior of sequences. Let’s see how it can help here:

• Input the first term \(a_1\) and the common difference \(d\) into the graphing utility.
• Generate the sequence terms up to the desired nth term, in this case, the tenth term.
• Compare the graphical representation or table with your calculated terms to verify accuracy.

By using a graphing utility, you make sure your manual calculations align with the visual data generated by the utility. It’s a convenient way to cross-check your work.

Sequence Terms

In an arithmetic sequence, **sequence terms** are the individual elements, each derived from adding the common difference to the previous term. In our example, terms start at 5, the next is 11, and we calculated that the 10th term is 59.

• First term \(a_1 = 5\)
• Second term \(a_2 = 11\)
• …
• Tenth term \(a_{10} = 59\)

Understanding how each term in a sequence is calculated and related can help deeply comprehend the pattern. Each new term builds logically on the last, establishing a clear, predictable pattern. This consistency is what makes arithmetic sequences so powerful and practical in both mathematics and real-life applications.