Problem 19 - 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 \(a_{200}\) when \(a_{1}=-40, d=5\)

Step-by-Step Solution

Answer

The 200th term of the arithmetic sequence is 955.

Step 1: Understand the provided values

In this problem, we are given the first term \(a_{1} = -40\), the common difference \(d = 5\), and the term number we need to find which is \(n = 200\).

Step 2: Apply the Arithmetic Sequence Formula

Apply the formula for the \(n\)th term of an arithmetic sequence: \(a_{n} = a_{1} + (n - 1)d\). Substitute the given values into the formula: \(a_{200} = -40 + (200 - 1) \cdot 5\).

Step 3: Compute the Result

Perform the operations: \(a_{200} = -40 + 199 \cdot 5 = -40 + 995 = 955\).

Key Concepts

Understanding the Arithmetic Sequence Formula

When studying arithmetic sequences, a critical tool is the arithmetic sequence formula, which allows us to find any term in a sequence. The formula \(a_n = a_1 + (n - 1)d\) is powerful in its simplicity, guiding us directly to the term we seek. Here, \(a_n\) represents the term of the sequence you want to find, \(a_1\) is the first term in the sequence, \(n\) is the term number, and \(d\) is the common difference between terms.

To use this formula, we need to identify our knowns: the first term and the common difference. Next, we plug these values into our formula, along with the position of the term we're looking for. This process provides a straightforward path to finding any term within the arithmetic sequence, making it an invaluable equation for solving such problems.

Deciphering the Common Difference

The concept of the common difference, denoted by \(d\), is the key characteristic of an arithmetic sequence. It refers to the steady increment or decrement that occurs from one term to the next. Every adjacent pair of terms in the sequence will have this same common difference when subtracting the preceding term from the following term.

For example, if we were working with a sequence where each term increases by 2, then our common difference \(d\) would be 2. This means if we start with a first term of 3, the sequence would progress as 3, 5, 7, and so on, each term adding 2—the common difference—to the previous term. Understanding \(d\) is crucial because it defines the pattern of the sequence and is essential for using the arithmetic sequence formula effectively.

Calculating a Specific Sequence Term

Let's look closely at the process of finding a specific term in an arithmetic sequence, known as sequence term calculation. By following the arithmetic sequence formula, \(a_n = a_1 + (n - 1)d\), we systematically approach this calculation. It's important to pay close attention to the values used for \(n\) and \(d\), as any errors here will lead to an incorrect sequence term.

In the context of an exercise, suppose we want to find the 50th term of a sequence that starts with 7 and increases by 3 each time. Plugging these values into our formula, we have \(a_{50} = 7 + (50 - 1) \times 3\). After doing the math, the 50th term is found to be 154. This step-by-step approach makes sequence term calculation approachable, ensuring that students can find any term quickly and with confidence.