Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the arithmetic sequence. Find the common difference and write the \(n\) th term of the sequence as a function of \(n .\) $$a_{1}=1.5, a_{k+1}=a_{k}-2.5$$

Step-by-Step Solution

Answer

The first five terms of the arithmetic sequence are \(1.5, -1.0, -3.5, -6.0, -8.5\). The common difference is -2.5. The nth term of the sequence as a function of \(n\) is \(a_n = 4 - 2.5n\).

Step 1: Find the first five terms of the sequence

To find the first term \(a_1\), it is already given in the exercise, that is \(a_1 = 1.5\). To find the second term \(a_2\), you need to substitute \(k = 1\) into the expression \(a_{k+1} = a_k - 2.5\), which gives \(a_2 = 1.5 - 2.5 = -1.0\). Similarly, for the third term, substitute \(k = 2\) to get \(a_3 = a_2 - 2.5 = -1.0 - 2.5 = -3.5\). Continue in this manner to get \(a_4 = -6.0\) and \(a_5 = -8.5\). So, the first five terms are \(1.5, -1.0, -3.5, -6.0, -8.5\).

Step 2: Find the common difference

The common difference \(d\) for an arithmetic sequence is the difference between consecutive terms. From the sequence terms found in step 1, \(d = a_2 - a_1 = -1.0 - 1.5 = -2.5\). So, the common difference is -2.5.

Step 3: Write the nth term of the sequence as a function of \(n\)

The nth term of the sequence can be found by substituting \(n\), \(a_1\) and the common difference \(d\) into the formula \(a_n = a_1 + (n-1) * d\). This gives the nth term as \(a_n = 1.5 + (n-1) * -2.5 = 4 - 2.5n\).

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, the "common difference" is the amount you add or subtract to get from one term to the next. It is a constant and plays a critical role in defining the sequence. When you look at an arithmetic sequence, you'll notice that this difference remains the same throughout.

For example, if the first term is 1.5 and the second term is -1.0, the common difference can be found by subtracting the first term from the second term. So,

• Second term - First term = Common difference
• \(-1.0 - 1.5 = -2.5\)

This tells us that every term is 2.5 less than the one before. Therefore, in this example, the common difference is -2.5. This value will allow us to find all following terms in the sequence.

What are Sequence Terms?

Sequence terms are the actual numbers that form the arithmetic sequence. You begin with the first term, known as \(a_1\), and use the common difference to find subsequent terms. When working through a problem that involves an arithmetic sequence, your first step is often to list these initial terms.

Given \(a_1 = 1.5\) and a common difference of -2.5, you can easily find the sequence terms by repeatedly subtracting the common difference from the preceding term. Initially, your sequence will look like this:

• First term (\(a_1\)): 1.5
• Second term (\(a_2\)): 1.5 - 2.5 = -1.0
• Third term (\(a_3\)): -1.0 - 2.5 = -3.5
• Fourth term (\(a_4\)): -3.5 - 2.5 = -6.0
• Fifth term (\(a_5\)): -6.0 - 2.5 = -8.5

By writing out these terms, you can better visualize how each term fits into the sequence.

How to Use the nth Term Formula

The "nth term formula" is a powerful tool in arithmetic sequences. It allows you to find any term in the sequence without having to list all the terms that come before it. The formula is given by:

• \(a_n = a_1 + (n-1) \times d\)

Here, \(a_1\) is the first term, \(n\) is the term number you want to find, and \(d\) is the common difference.

For the sequence provided with \(a_1 = 1.5\) and \(d = -2.5\), plug these values into the nth term formula as follows:

• \(a_n = 1.5 + (n-1) \times (-2.5)\)
• Simplifying further, we get \(a_n = 4 - 2.5n\)

This formula is versatile and can be used to calculate any term, making it a valuable method to quickly access any part of the sequence.