Problem 19 - College Algebra

Question

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: Identify the given information

First, identify and write down the values provided in the prompt. This gives: first term, \(a_1 = -40\), common difference, \(d = 5\), term position, \(n = 200\).

Step 2: Apply the formula for the \(n\)th term of an arithmetic sequence

Now, plug these values into the formula for the \(n\)th term in an arithmetic sequence. Using the formula \(a_n = a_1 + (n-1) * d\), we get: \(a_{200} = -40 + (200-1) * 5\).

Step 3: Calculate the 200th term of the sequence

Perform the calculation given from the formula. \(a_{200} = -40 + 199 * 5 = -40 + 995 = 955\).

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, each term is derived from the previous one by adding a fixed number, known as the common difference. This number remains constant throughout the sequence. For example, if the first term \(a_1\) is -40 and the common difference \(d\) is 5, the sequence begins as -40, -35, -30, and so on.

Here is why the common difference is critical:

• It determines the spacing between terms in the sequence.
• A positive value means the sequence increases, while a negative value indicates it decreases.
• It's found by subtracting any term from its succeeding term.

This concept ensures predictability, allowing us to calculate any term swiftly.

Using the nth Term Formula

The nth term formula is essential for determining any term in an arithmetic sequence without listing all the terms. It is given by: \[a_n = a_1 + (n-1) \times d\] Here:

• \(a_n\) represents the nth term we aim to find.
• \(a_1\) is the first term of the sequence.
• \(n\) stands for the term number we're interested in.
• \(d\) is the common difference.

Using this formula, you can jump directly to any term. For instance, to find \(a_{200}\), plug in \(a_1 = -40\), \(n = 200\), and \(d = 5\) to get \(a_{200} = -40 + (200-1) \times 5\).

It provides a shortcut, saving time and effort.

Carrying Out Sequence Calculation

Once the formula is set, the next step is to perform the arithmetic calculations needed. This involves substituting in the known values and simplifying. For our example sequence:

• Start with the formula: \(a_{200} = -40 + (200-1) \times 5\)
• Calculate the expression inside the parentheses: 199
• Multiply 199 by 5 to get 995
• Add this product to the first term: -40 + 995
• The result, \(a_{200} = 955\), is your term.

Each step builds on the understanding of arithmetic operations.

This process not only reinforces arithmetic skills but also illustrates the power of the nth term formula in effortlessly finding any required term.