Problem 24 - A Graphical Approach to College Algebra

Question

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{4}=e+2 \sqrt{\pi}, a_{5}=e+3 \sqrt{\pi}$$

Step-by-Step Solution

Answer

\(a_8 = e + 6\sqrt{\pi}\) and \(a_n = e + (n-2)\sqrt{\pi}\).

Step 1: Identify Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant. Here, the common difference \(d\) can be calculated as: \[ d = a_5 - a_4 = (e + 3 \sqrt{\pi}) - (e + 2 \sqrt{\pi}) = \sqrt{\pi}. \]

Step 2: Use the Arithmetic Sequence Formula

The general formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)\cdot d\). However, since \(a_1\) is not directly given, we first determine \(a_1\) using the known terms by assuming a value for \(a_n\) such as \(a_4\).

Step 3: Solve for \(a_1\)

Use the formula for the 4th term: \[ a_4 = a_1 + 3d = e + 2\sqrt{\pi}. \] Plugging in \(d = \sqrt{\pi}\), we get \[ a_1 + 3 \sqrt{\pi} = e + 2\sqrt{\pi}. \] Thus, \[ a_1 = e + 2\sqrt{\pi} - 3\sqrt{\pi} = e - \sqrt{\pi}. \]

Step 4: Calculate \(a_8\)

Now that we have \(a_1 = e - \sqrt{\pi}\) and \(d = \sqrt{\pi}\), we can find \(a_8\) using the formula: \[ a_8 = a_1 + 7d = (e - \sqrt{\pi}) + 7 \cdot \sqrt{\pi} = e - \sqrt{\pi} + 7\sqrt{\pi} = e + 6\sqrt{\pi}. \]

Step 5: Express the General Term \(a_n\)

Using \(a_1\) and \(d\), the general formula for the \(n\)-th term is: \[ a_n = a_1 + (n-1)\cdot d = (e - \sqrt{\pi}) + (n-1)\cdot \sqrt{\pi} = e + (n-2)\sqrt{\pi}. \]

Key Concepts

Common Difference

The common difference is a fundamental component of arithmetic sequences. It's the consistent difference between consecutive terms, driving the sequence forward.
In the given example, the common difference \(d\) was identified between the 4th and 5th terms:

• \(a_4 = e + 2\sqrt{\pi}\)
• \(a_5 = e + 3\sqrt{\pi}\)

The difference is calculated as:\[d = a_5 - a_4 = (e + 3\sqrt{\pi}) - (e + 2\sqrt{\pi}) = \sqrt{\pi}.\] Recognizing this difference is crucial because it allows us to predict future terms and understand the sequence better.

Arithmetic Sequence Formula

The arithmetic sequence formula is used to find any term in the sequence. It's typically expressed as:\[a_n = a_1 + (n-1) \cdot d\]where:

• \(a_n\) is the \(n\)-th term.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.

In our exercise, we needed to determine \(a_1\) first. By given terms \(a_4\) and \(a_5\), we back-calculated \(a_1\) using:\[a_4 = a_1 + 3 \cdot d = e + 2\sqrt{\pi}.\]This formula is particularly useful because once you have \(a_1\) and \(d\), you can find any sequence term no matter how large \(n\) is.

General Term of a Sequence

The general term of a sequence provides a handy way to express any term \(a_n\) in the sequence without needing explicit calculation for each term. With the equation \[a_n = a_1 + (n-1)\cdot d\] from the arithmetic sequence formula, every term can be found directly by modifying the term's index \(n\). By substituting \(a_1 = e - \sqrt{\pi}\) and \(d = \sqrt{\pi}\), we derived:\[a_n = (e - \sqrt{\pi}) + (n-1)\sqrt{\pi} = e + (n-2)\sqrt{\pi}.\] This expression allows quick reference to any term's value just by plugging the value of \(n\). Useful for arithmetic patterns, this method smoothens the understanding of sequence behavior and application.