Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, find the number of terms in the given finite arithmetic sequence. \(a=\\{3,-4,-11, \ldots,-60\\}\)

Step-by-Step Solution

Answer

The sequence has 10 terms.

Step 1: Identify the first term and the common difference

The given arithmetic sequence is \(3, -4, -11, \ldots, -60\). The first term \(a_1\) is 3. To find the common difference \(d\), subtract the first term from the second term: \(-4 - 3 = -7\). Therefore, \(d = -7\).

Step 2: Set up the arithmetic sequence formula

The 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 of the sequence, \(a_1\) is the first term, and \(d\) is the common difference.

Step 3: Solve for the number of terms

We need to find the number of terms \(n\) for which the \(n\)-th term is \(-60\). Substitute \(a_n = -60\), \(a_1 = 3\), and \(d = -7\) into the sequence formula: \(-60 = 3 + (n-1)(-7)\).

Step 4: Simplify the equation

Solve \(-60 = 3 + (n-1)(-7)\) to find \(n\). First, simplify the equation: \(-60 = 3 - 7(n-1)\). Then, simplify further: \(-60 = 3 - 7n + 7\).

Step 5: Further simplify and solve for \(n\)

Combine like terms: \(-60 = 10 - 7n\). Subtract 10 from both sides: \(-70 = -7n\). Divide by -7 to solve for \(n\): \(n = 10\).

Key Concepts

Understanding Finite Sequences

A finite sequence is a list of numbers that has a definite starting and ending point. This means it contains a specific number of terms. Finite sequences are different from infinite sequences, which continue indefinitely without stopping. The task or exercise you are working on is dealing with a specific finite arithmetic sequence.
This particular sequence starts with the number 3 and ends with the number -60.
Because it is finite, we can actually count how many numbers or terms are in the sequence.

• Finite sequences are important because they help us understand patterns within a fixed range.
• In this case, knowing the sequence is finite allows us to determine the number of terms.

By determining the elements of the sequence, you can gain a better understanding of arithmetic progressions and how they function in a closed system.

Exploring the Common Difference

In an arithmetic sequence, the common difference is the amount by which each term increases or decreases to reach the next term.
This difference remains constant throughout the sequence. You calculate it by subtracting any term from the term that follows it.
In our example, the sequence is 3, -4, -11, ..., -60. We found that the common difference, denoted as \( d \), is -7. Here is how:

• Take the second term, -4, and subtract the first term, 3.
• The result, \( -4 - 3 = -7 \), is our common difference.

Knowing the common difference tells us how the sequence progresses.
This term sequence decreases by 7 each step.
Understanding this concept helps unravel the arithmetic pattern.

Identifying the First Term

The first term of an arithmetic sequence is the initial value from which we start.
It is often denoted by \( a_1 \). In this exercise, the first term is 3.

• The first term sets the stage for all subsequent terms since each one is calculated by adding the common difference to it.
• Identifying the first term is crucial because it anchors the entire sequence.

Once you know the first term and the common difference, you can calculate any term within the sequence.
Such information is vital when applying the arithmetic sequence formula to solve exercises or understand the progression.

Calculating the Number of Terms

To find the number of terms in the finite sequence, we use the formula for the \( n \)-th term of the arithmetic sequence: \[ a_n = a_1 + (n-1) imes d \] Here, \( a_n \) is the last term, \( a_1 \) is the first term, and \( d \) is the common difference. In our sequence, \( a_n = -60 \), \( a_1 = 3 \), and \( d = -7 \). By substituting these values into the formula, we get: \[ -60 = 3 + (n-1) imes (-7) \] From this, we can solve for \( n \):

• Simplify to find \(-60 = 3 + (-7n + 7)\)
• Rearrange to \(-60 = 10 - 7n\)
• Solve for \( n \) to find \( n = 10 \)

Understanding how to manipulate the arithmetic sequence formula enables us to find not just the number of terms, but also specific terms if needed.
This step is essential when working with finite arithmetic sequences.