Problem 18 - College Algebra

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\). Find \(a_{60}\) when \(a_{1}=8, d=6\)

Step-by-Step Solution

Answer

After the calculations in the last step, the 60th term of the arithmetic sequence, \(a_{60}\), is \(362\).

Step 1: Identify the Values

First, identify the first term, \(a_{1}\), and the common difference, \(d\), from the given question. In our case, \(a_{1} = 8\) and \(d = 6\). We are also given that we need to find the 60th term (\(n=60\)).

Step 2: Apply the Formula

Apply the formula for the nth term in an arithmetic sequence, which is \(a_{n}= a_{1} + (n - 1) * d\). The first term (\(a_{1}\)) is known to us as \((8)\), the common difference (\(d\)) is \((6)\), and \(n\) represents the term number that we are trying to find, in this case \((60)\). This will give us \(a_{60}= a_{1} + (60 - 1) * d\).

Step 3: Calculate

Now, substitute \(a_{1}\) and \(d\) with \(8\) and \(6\) into the formula, so we get \(a_{60}= 8 + (60 - 1) * 6\). Now we need to follow the order of operations and calculate the result.

Key Concepts

nth term formula

The "nth term formula" in an arithmetic sequence is a handy tool that allows you to find any term in a sequence, simply by knowing the first term and the common difference. It’s a very straightforward formula:

• \( a_{n} = a_{1} + (n-1) \times d \)

Where:

• \( a_{n} \) is the nth term you want to find.
• \( a_{1} \) is the first term in the sequence.
• \( n \) is the term number you’re interested in.
• \( d \) is the common difference between consecutive terms.

To apply this formula, substitute the known values of \(a_{1}\), \(d\), and \(n\) to find \(a_{n}\). For example, in our exercise, finding \(a_{60}\) involves plugging the first term \(a_{1} = 8\), common difference \(d = 6\), and \(n = 60\) into this formula.

first term

The "first term" in an arithmetic sequence is crucial because it serves as the starting point for the entire sequence. Think of it as the seed from which the sequence grows. Every term that follows is built upon this first term by adding the common difference.

• The first term is represented by \( a_{1} \).

In many situations, you will be given the first term, as in our example where \( a_{1} = 8 \). The first term is important for calculating any other term in the sequence using the nth term formula. Hence, knowing the first term sets the foundation for solving and understanding the entire arithmetic sequence.

common difference

The "common difference" in an arithmetic sequence dictates how much each term increases from the previous one. Essentially, this value determines the rate of change within the sequence. It's symbolized by \( d \), and its value remains constant throughout the arithmetic sequence.

• \( d = a_{n+1} - a_{n} \)

To find the common difference, subtract any term from the subsequent term in the series. In our example, the common difference \( d \) is 6. This means each term is 6 more than the term before it.Understanding the common difference helps in predicting future terms and easily generating a sequence from its starting point.