Problem 47 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the arithmetic sequence. $$a_{1}=8, a_{4}=17$$

Step-by-Step Solution

Answer

The general term is \(a_n = 3n + 5\).

Step 1: Understand the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This difference is called the common difference, denoted as \(d\).

Step 2: Identify the Given Values

From the problem, we know that the first term \(a_1 = 8\) and the fourth term \(a_4 = 17\).

Step 3: Use the Formula for the nth Term

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference.

Step 4: Calculate the Common Difference

Since \(a_4 = a_1 + 3d = 17\), we substitute \(a_1 = 8\) into the equation: \(8 + 3d = 17\). Solve for \(d\): \[3d = 17 - 8 = 9 \]\[d = \frac{9}{3} = 3 \]

Step 5: Write the General Term Formula

Substitute \(a_1 = 8\) and \(d = 3\) into the general term formula \(a_n = a_1 + (n-1)d\). \[a_n = 8 + (n-1) \times 3 = 8 + 3n - 3\]Simplify:\[a_n = 3n + 5\]

Step 6: General Term Verification

Verify by checking some terms:- \(a_1 = 3 \cdot 1 + 5 = 8\), which is correct.- \(a_4 = 3 \cdot 4 + 5 = 12 + 5 = 17\), which is also correct.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a very important value. This is the consistent difference between each pair of consecutive terms in the sequence. For example, if each number in the sequence is larger than the previous one by 3, then the common difference, denoted by \(d\), is 3.

• The common difference can be found by subtracting one term from another. Given two terms \(a_n\) and \(a_m\), \(d = a_m - a_n\).
• In our problem, we know \(a_1 = 8\) and \(a_4 = 17\). The distance from \(a_1\) to \(a_4\) is three steps (think of this as hops or jumps).
• To find \(d\), we can set up the equation: \(8 + 3d = 17\). Solving this gives us \(d = 3\).

Understanding the common difference allows you to see the pace at which the sequence increases or decreases. It essentially builds a "recipe," showing how each subsequent term in the sequence is derived.

General Term Formula

Once you have determined the common difference, you can find the general term formula for the sequence. This formula allows you to find any term in the arithmetic sequence without having to list all the previous ones.

• The formula used is \(a_n = a_1 + (n-1)d\).
• \(a_1\) is the initial term, which is 8 in our example.
• \(d\) is the common difference previously calculated as 3.

Applying these values in the formula gives us \(a_n = 8 + (n-1) \times 3\).
Simplifying, you get \(a_n = 3n + 5\).
This equation can be used to find any term "\(n\)" in our sequence swiftly, without recalculating the terms step by step.

Sequence Verification

Verification ensures that the general term formula accurately represents the sequence. To verify, we calculate certain terms using the formula and compare these calculated values with known terms.

• Start with \(a_1 = 3 \cdot 1 + 5 = 8\), matching it with the first term provided.
• Check \(a_4 = 3 \cdot 4 + 5 = 12 + 5 = 17\), confirming it matches the given fourth term.

During verification, going through some terms of the sequence can enhance understanding. It ensures no errors have been made, and the formula aligns perfectly with the real sequence information. Effective verification further consolidates the sequence and the math behind your solution, instilling confidence in the obtained result.