Problem 21 - 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 \(a_{1}=35, d=-3\)

Step-by-Step Solution

Answer

The 60th term of the sequence is -142.

Step 1: Identify the given values

First, identify the given values in the exercise. Here, \(a_{1}=35\) which is the first term of the sequence, \(d=-3\) which is the common difference, and we need to find \(a_{60}\), the 60th term of the sequence.

Step 2: Apply the formula of an arithmetic sequence

The formula of the nth term of an arithmetic sequence is \(a_{n}=a_{1}+(n-1)d\). Plug in the given values into this formula: \(a_{60}=a_{1}+(60-1)d\).

Step 3: Simplify the expression

After plugging in the values, simplify the expression to obtain the value of \(a_{60}\). So, \(a_{60}=35+(60-1)(-3)=35+(-177)=-142\).

Key Concepts

First Term

The first term of an arithmetic sequence is the initial number from which the sequence begins. In any arithmetic sequence, the first term is denoted by \(a_1\). This term serves as the foundational block that defines the entire sequence. In the context of our exercise, the first term \(a_1\) is given as 35.
The first term is crucial because it sets the stage for how the sequence progresses. Subsequent terms are computed based on this initial term, adding or subtracting the common difference repeatedly. Imagining the first term as the starting point helps visualize the linear pattern the sequence will follow.

• Begin with the first term, \(a_1\).
• Add the common difference \(d\) successively to find subsequent terms.
• Continue this process to find the nth term.

Understanding the first term is essential to solving arithmetic sequences and decoding the pattern it follows. Once you grasp this, finding any term in the sequence becomes straightforward.

Common Difference

The common difference in an arithmetic sequence is a constant value that defines the change between consecutive terms. It is represented by \(d\) and can be either positive or negative.
In the exercise provided, the common difference \(d\) is -3. This means that to determine the next term in the sequence, you subtract 3 from the current term. A negative common difference implies that the sequence is decreasing with each step.
The concept of common difference is critical because it dictates the direction and rate at which the sequence grows or shrinks. Here's how it works:

• Start with the first term, \(a_1\).
• Add the common difference \(d\) continuously to find the next terms.
• A positive \(d\) means the sequence is increasing, while a negative \(d\) means it is decreasing.

By understanding and applying the common difference, you can easily reconstruct the sequence and predict future terms.

nth Term Formula

The nth term formula is a mathematical expression used to find any term in an arithmetic sequence without listing all preceding terms. It is represented by \(a_n = a_1 + (n-1)d\), where:

• \(a_n\) represents the nth term.
• \(a_1\) is the first term.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference.

In our exercise, the nth term formula is used to find \(a_{60}\), the 60th term. By substituting the known values, \(a_1 = 35\), \(n = 60\), and \(d = -3\), into the formula, you perform the calculation:\[a_{60} = 35 + (60-1)(-3) = 35 + (-177) = -142\]The nth term formula is powerful because it allows direct calculation of any term in the sequence, saving time and effort. This formula is a key tool in arithmetic sequences, enabling a clear and efficient method to find specific terms.