Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{1}=-2.6, d=0.2$$

Step-by-Step Solution

Answer

The first five terms of the sequence are -2.6, -2.4, -2.2, -2.0, and -1.8.

Step 1: Identify the first term and common difference of the sequence

The first term \(a_{1}\) of the sequence is given as -2.6 and the common difference \(d\) is 0.2. This information was provided in the problem statement.

Step 2: Calculate the second term of the sequence

To find the second term (\(a_{2}\)) of the sequence, add the common difference to the first term. So, \(a_{2} = a_{1} + d = -2.6 + 0.2 = -2.4\)

Step 3: Calculate the third term of the sequence

Similarly, to find the third term (\(a_{3}\)), add the common difference to the second term: \(a_{3} = a_{2} + d = -2.4 + 0.2 = -2.2\)

Step 4: Calculate the fourth term of the sequence

Do the same thing for the fourth term (\(a_{4}\)): \(a_{4} = a_{3} + d = -2.2 + 0.2 = -2.0\)

Step 5: Calculate the fifth term of the sequence

Lastly, find the fifth term (\(a_{5}\)) by adding the common difference to the fourth term: \(a_{5} = a_{4} + d = -2.0 + 0.2 = -1.8\)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a critical element that determines the pattern of the sequence. It's the constant value we add to each term to get to the next one. Think of it as a step size in your sequence. For any sequence, this difference is denoted by the letter \(d\). In our example, \(d = 0.2\). This means that each subsequent term is found by adding 0.2 to the previous term:- Start from \(a_{1} = -2.6\)- To find \(a_{2}\), add 0.2: \(-2.6 + 0.2 = -2.4\)- To find \(a_{3}\), add 0.2: \(-2.4 + 0.2 = -2.2\)This small but essential number, 0.2 in this case, guides us in generating the sequence, showing how each term relates to the one before it. Without this value, we wouldn't know how the sequence continues.

Graphing Utility

A graphing utility is a powerful tool that helps visualize numerical patterns and sequences. In the context of an arithmetic sequence, it can be used to confirm that the sequence is consistent and correctly calculated. By plugging your arithmetic sequence into a graphing calculator, you immediately see the pattern visualized. This visual representation helps identify if the sequence is increasing, decreasing, or staying constant. Using a graphing utility involves:

• Inputting the sequence formula or manual terms into the graphing table.
• Checking that each plotted point aligns linearly, as arithmetic sequences are linear.
• Ensuring that calculations were performed correctly, confirmed by observing the straight line formed by sequence terms on the graph.

In our case, entering \(-2.6\), \(-2.4\), \(-2.2\), \(-2.0\), and \(-1.8\) into a graphing utility provides a straightforward way to verify these terms visually.

First Term

The first term of an arithmetic sequence is where it all begins. It's the starting point from which all other terms are derived by continuously adding the common difference. In mathematical terms, this first term is denoted by \(a_{1}\). In our example, \(a_{1} = -2.6\). Here's why the first term is important:

• It establishes the initial value for the entire sequence.
• Each subsequent term builds directly upon this initial number.
• Altering this single number can change the entire sequence dramatically.

Thus, correctly identifying the first term is crucial for accurately writing out the rest of the arithmetic sequence. Remember, once you know \(a_{1}\) and the common difference, the entire sequence follows smoothly from these two pieces of information.