Problem 59 - Algebra 2

Question

Find the indicated term of each arithmetic sequence. \(a_{43}\) for \(5,9,13,17, \ldots\)

Step-by-Step Solution

Answer

The 43rd term is 173.

Step 1: Identify the First Term

In an arithmetic sequence, each term after the first is found by adding a constant to the previous term. Here, the first term \(a_1\) is \(5\).

Step 2: Determine the Common Difference

The common difference \(d\) is the difference between any two consecutive terms. For the sequence \(5, 9, 13, 17, \ldots\), subtract the first term from the second: \(d = 9 - 5 = 4\).

Step 3: Use the Arithmetic Sequence Formula

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

Step 4: Plug in the Values

To find \(a_{43}\), substitute \(a_1 = 5\), \(d = 4\), and \(n = 43\) into the formula:\[ a_{43} = 5 + (43-1) \times 4 \]

Step 5: Simplify the Expression

First, calculate \(43 - 1\) which is \(42\). Then multiply by \(4\):\[ 42 \times 4 = 168 \]Now add to the first term: \(5 + 168 = 173\).

Step 6: Conclude the Solution

Therefore, the 43rd term of the arithmetic sequence is \(173\).

Key Concepts

Common Difference

A key concept of arithmetic sequences is the common difference, which is constant between consecutive terms. It represents the value that you add to each term to get to the next one.
Understanding the common difference helps you predict the growth of the sequence. It is determined by subtracting one term from its preceding term. In the sequence \(5, 9, 13, 17, \ldots\), the common difference \(d\) is calculated as \(d = 9 - 5 = 4\).
This consistency is what makes arithmetic sequences predictable and easy to work with.

nth Term Formula

The nth term formula is a tool used to find any term in an arithmetic sequence without listing all the terms. It's an equation that relates a term's position in the sequence to its value.
The formula is given by: \[ a_n = a_1 + (n-1) \, d \]Here, \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the number of the term you want to find.
The formula simplifies your work and helps in calculating distant terms like the 43rd term in the sequence.

Sequence Term Calculation

Calculating a specific term in an arithmetic sequence requires substituting known values into the nth term formula. It eliminates the need to sequentially add the common difference multiple times.
For the sequence \(5, 9, 13, 17, \ldots\), to find \(a_{43}\), use the nth term formula \(a_{43} = a_1 + (n-1)\, d\).
Substituting the values \(a_1 = 5\), \(d = 4\), and \(n = 43\), you calculate:

• \(a_{43} = 5 + 42 \times 4\)
• \(a_{43} = 5 + 168\)
• \(a_{43} = 173\)

The 43rd term is \(173\). This approach is efficient and mistake-free compared to adding repeatedly.

Arithmetic Sequence Formula

The arithmetic sequence formula is central to understanding arithmetic sequences. It reflects how each term can be generated from its position and existing sequence attributes.
This formula is \(a_n = a_1 + (n-1) \, d\), encapsulating all you need to calculate any specific term. By having precise control over the term's position \(n\), it becomes straightforward to figure out what its value will be.
The arithmetic sequence formula embodies the sequence’s structure and logic, providing a shortcut to any term you want to find without exhaustive calculations.