Problem 2 - Algebra 2 and Trigonometry

Question

Pedro said that to form a sequence of five terms that begins with 2 and ends with \(12,\) you should divide the difference between 12 and 2 by 5 to find the common difference. Do you agree with Pedro? Explain why or why not.

Step-by-Step Solution

Answer

No, Pedro is incorrect; the correct common difference is 2.5, not 2.

Step 1: Understanding the Problem

Pedro suggests that to form an arithmetic sequence starting at 2 and ending at 12 with five terms, we should divide the difference between 12 and 2 by 5 to find the common difference.

Step 2: Calculate the Overall Difference

First, we calculate the difference between the starting and ending terms of the sequence: \[ 12 - 2 = 10 \]

Step 3: Common Difference Formula Setting

An arithmetic sequence with 5 terms has 4 equal intervals (common differences) between the initial and final term. Therefore, the common difference should be calculated as: \[ \text{Common Difference} = \frac{10}{4} = 2.5 \]

Step 4: Evaluate Pedro's Method

Pedro suggested using 5 as the divisor, but since the sequence has 4 intervals, the correct divisor is 4, not 5.

Step 5: Conclusion

Pedro's method is incorrect because he used the wrong number of intervals to determine the common difference. The correct common difference should be 2.5, not 2.

Key Concepts

Understanding the Common Difference

The concept of 'common difference' is crucial in understanding arithmetic sequences. An arithmetic sequence is a list of numbers where the difference between any consecutive terms is always the same. This repeated difference is called the 'common difference'. For instance, if our sequence started at 2 and the common difference was 2, our sequence would look like this: 2, 4, 6, 8, and so on.
The formula to find the common difference is simple: Subtract the previous term from the next term. It helps in predicting future terms in the sequence and understanding the pattern that the sequence follows.

Sequence Terms Explained

In any sequence, each number is referred to as a 'term'. Understanding sequence terms is fundamental in navigating arithmetic sequences. Let's say we have an arithmetic sequence with terms represented as: 2, 4, 6, 8, 10. Here, the numbers 2, 4, 6, 8, and 10 are the terms of the sequence.
It's important to note that the terms are ordered, and the sequence can go on infinitely if continued. In the exercise, there are five terms in total, including the starting and ending numbers.

Intervals in Arithmetic Sequences

Intervals in arithmetic sequences refer to the 'gaps' or 'steps' between each consecutive term. These intervals or steps are equal in an arithmetic sequence. This is why, although there are 5 terms in Pedro's sequence, there are only 4 intervals, because the interval is the space between two terms.
Calculating intervals is a critical step in finding the common difference. If the sequence terms start at 2 and end at 12, the intervals are evenly distributed across the sequence. Thus, dividing the total difference (12-2) by the number of these intervals (4) gives the common difference.

Understanding the Arithmetic Progression Error

Arithmetic progression errors occur when there is a misunderstanding of the underlying principles of arithmetic sequences. In the exercise example, Pedro made an arithmetic progression error by incorrectly calculating the number of intervals.
He divided by 5 instead of the correct 4 (the number of gaps between terms) to determine the common difference. This resulted in an incorrect common difference implying an error in the sequence formation. Recognizing and correcting such mistakes is important to accurately solve and comprehend arithmetic sequence problems.