Problem 18 - College Algebra

Question

In Exercises \(15-22,\) 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

The 60th term of the arithmetic sequence is 362.

Step 1: Identify the given values

The given values are:First term: \(a_{1} = 8\)Common difference: \(d = 6\)Term to find: \(n = 60\)

Step 2: Substitute the given values into the formula for the nth term of an arithmetic sequence

Now, substitute the given values into the formula \(a_{n} = a_{1} + (n-1)d\). This gives:\(a_{60} = 8 + (60-1)6\)

Step 3: Calculate the value of \(a_{60}\)

To calculate \(a_{60}\), perform the operations:\(a_{60} = 8 + 59*6\)\(a_{60} = 8 + 354\)\(a_{60} = 362\)

Key Concepts

Arithmetic Sequence Formula

To comprehend an arithmetic sequence, it is crucial to understand its defining feature - a common difference between consecutive terms. In an arithmetic sequence, you can calculate any term using a straightforward formula:
\[ a_n = a_1 + (n - 1)d \]
where \( a_n \) is the nth term you’re looking for, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) represents the term number.

Why is this formula important?

This formula is powerful because it encapsulates the essence of an arithmetic sequence in a concise manner, permitting the calculation of any term based on its position without having to list out all the previous terms. For instance, to find the 60th term in a sequence, you don't need to know the 59th or any other specific term; you just need the first term and the common difference.

Common Difference

The common difference in an arithmetic sequence is like the heartbeat of the sequence - a consistent rhythm that defines its progression. Each step in the sequence increases by this set amount. In technical terms, the common difference (denoted as \( d \)) is the difference between any two successive terms.

Unearthing the Common Difference

To reveal the common difference, simply subtract one term from the following term. For example, if you have a sequence where subsequent terms are 3, 7, 11, then the common difference is \( 7 - 3 = 4 \). Knowing the common difference is critical because it allows us to predict the future terms of the sequence, or even backtrack to previous terms.

Sequence Term Calculation

For students tackling arithmetic sequences, the method of calculating any specific term (such as \( a_{60} \)) can sometimes seem daunting. Yet, with the right knowledge and formula, it's like following a map to a treasure. After identifying the first term and the common difference, you insert these values into our previously discussed formula to pinpoint the exact value of any term in the sequence.
By practicing sequence term calculation, students not only grasp the overall concept more firmly, but also become adept at predicting patterns - a skill that is beneficial beyond mathematics in many real-world scenarios.