Problem 15 - Prealgebra

Question

Find the next term in each list. \(67,61,55,49,43, \dots\)

Step-by-Step Solution

Answer

The next term in the sequence is 37.

Step 1: Identify the Pattern

Observe the sequence: 67, 61, 55, 49, 43, ... and note the difference between consecutive terms. Calculate the difference: 67 - 61 = 6, 61 - 55 = 6, 55 - 49 = 6, and 49 - 43 = 6. The difference between each consecutive term is consistently 6.

Step 2: Determine the Rule

The pattern identified in Step 1 is that the sequence decreases by 6 each time. This suggests that the sequence follows an arithmetic pattern where each term is obtained by subtracting 6 from the previous term.

Step 3: Calculate the Next Term

Use the rule identified in Step 2 to find the next term in the sequence. Take the last term, 43, and subtract 6: 43 - 6 = 37.

Key Concepts

Patterns in Sequences

Recognizing patterns is an important part of understanding sequences. In arithmetic sequences, like the one given here, each term is connected by a common difference. This means that the difference between any two consecutive terms is always the same.
To spot the pattern in a sequence, begin by calculating the difference between each term. In our example:

• The difference between 67 and 61 is 6.
• The difference between 61 and 55 is also 6.
• This consistent difference of 6 between the terms means that you are observing an arithmetic sequence.

Recognizing this uniformity helps in identifying the rule that governs the sequence.

Finding the Next Term

Once you have identified the consistent pattern in an arithmetic sequence, finding the next term becomes straightforward. You apply the rule of the sequence to the last term.
For our sequence, the rule is to subtract 6 from the current term to find the following term:

• You start with the last term listed, which is 43.
• Apply the rule by subtracting 6 from 43.
• This calculation results in the next term: 37.

By understanding how to apply the sequence's rule, you can extend the sequence to find future terms.

Step-by-Step Solutions

Understanding sequences is often easier when you break down the process into clear, logical steps. Here’s how we can approach finding the next term in a sequence, using the step-by-step solution:
1. **Identify the Pattern** - Start by calculating the difference between consecutive terms. Recognition of a consistent difference tells us this is an arithmetic sequence.
2. **Determine the Rule** - Use the pattern identified to formulate a rule. For the given sequence, the rule is to subtract 6 from each term to find the next one.
3. **Calculate the Next Term** - With the rule at hand, apply it to the last term to find the next term in the sequence. This last step shows the power of step-by-step solutions for visualizing problems and arriving at the answer, piece by piece, making complex exercises simpler to handle.