Problem 44 - Algebra 2

Question

a. Use your calculator to generate an arithmetic sequence with a common difference of - \(7 .\) How could you se a calculator to find the 6th term? The 8th term? The 20th term? b. Explain how your answer to part (a) relates to the explicit formula \(a_{n}=a_{1}+(n-1) d\)

Step-by-Step Solution

Answer

Using a calculator, we can generate an arithmetic sequence with a common difference of -7 by starting with 0 and subtracting 7 to generate each subsequent term. This same output could be achieved through the formula \(a_{n}=a_{1}+(n-1) d\), as both work to add or subtract the common difference from the first term to find the nth term.

Step 1: Generating the sequence

Begin by inputting the sequence's first term to your calculator. For this demonstration, let's assume our first term is 0. By pressing the + key, the calculator remembers this operation. We can now subtract 7 to find the next term in the sequence. Continue this process to find each subsequent term.

Step 2: Finding specific terms

To find the 6th, 8th, and 20th terms, calculate up to these numbers. The 6th term will require five subtractions of 7, the 8th will require seven, and the 20th will require nineteen.

Step 3: Correlating our findings with the formula

The formula \(a_{n}=a_{1}+(n-1) d\) denotes the nth term of an arithmetic sequence. In this equation, \(a_1\) is the first term, d is the common difference, and n is the term number. Our sequence started at 0 with a common difference of -7. Therefore, our formula for any term is \(a_{n} = 0 + (n-1)(-7)\), or abbreviated: \(a_{n} = -7n + 7\). Therefore, this formula will output the same results as our calculator.

Key Concepts

Common Difference

An arithmetic sequence is a list of numbers where each term after the first is derived by adding a fixed, constant number, known as the "common difference," to the previous term. Essentially, it's the consistent number you either add or subtract from one term to find the next.
If you're dealing with a sequence like the one described with a difference of -7, it means you subtract 7 from each term to obtain the subsequent term. This concept is foundational in understanding arithmetic sequences as it dictates the pattern of the sequence.
Here are key points about common difference:

• Positive common difference increases the sequence.
• Negative common difference, as we have it here, decreases the sequence.
• The common difference remains constant throughout the sequence.

Explicit Formula

The explicit formula for an arithmetic sequence lets you find any term without needing to list all previous terms. It's given by the formula: \[a_{n} = a_{1} + (n-1)d\]In this formula, \(a_{n}\) is the nth term you want to find, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. By plugging the proper values into this formula, you can jump directly to any term, like the 6th, 8th, or the 20th, in the sequence.
For our sequence, starting at 0 and a common difference of -7, the formula becomes \(a_{n} = 0 + (n-1)(-7)\), which simplifies to \(a_{n} = -7n + 7\). This shows how each term is a solution derived from a direct calculation.

Calculator Usage

Using a calculator can be a great way to understand arithmetic sequences through step-by-step repetition without manually writing every step. Begin by inputting the first term of your sequence. After that, choose the operation based on the common difference:

• If it's a positive difference, add that number.
• If it's a negative difference, like -7 in this example, subtract this number.

For finding the 6th, 8th, and 20th terms, you continue pressing the addition (or subtraction) key repeatedly until you reach the desired term number. Calculate easily using:

• First, enter the initial term.
• Then apply the operation for the given common difference.
• Continue the operation until reaching the target term.

This method provides a practical understanding of the structure of arithmetic sequences, while the explicit formula offers a more streamlined approach.

Term Calculation

The process of term calculation involves using either the repetitive approach with a calculator or the swift explicit formula to find specific terms in an arithmetic sequence. Let's explore each method:

Calculator Method:

For example, if you wish to find the 6th term with a common difference of -7 starting at 0, you repeatedly subtract 7 six times:

• Starting at term 1: 0,
• Term 2: 0 - 7 = -7,
• Term 3: -7 - 7 = -14,
• Continue until Term 6: -35.

Formula Method:

Using the explicit formula \(a_{n} = -7n + 7\):

• For the 6th term (\(n = 6\)): \(-7 \times 6 + 7 = -35\).
• For the 8th term (\(n = 8\)): \(-7 \times 8 + 7 = -49\).
• For the 20th term (\(n = 20\)): \(-7 \times 20 + 7 = -133\).

These approaches allow you to understand and calculate terms efficiently, giving a choice between a manual or formulaic approach depending on your preference.