Problem 73 - Algebra 1: Concepts and Skills

Question

List the next three numbers suggested by the sequence. (Skills Review pp. 781) $$ 60,57,53,48, ?, ? ,? $$

Step-by-Step Solution

Answer

Therefore, the next three numbers in the sequence are 42, 35, and 27.

Step 1: Identifying the Pattern

Observe the given sequence: \(60, 57, 53, 48\). By subtracting each subsequent number from the previous one, it's noticeable that the difference between numbers increases by 1 each time.

Step 2: Applying the Identified Pattern

The first difference is 60 - 57 = 3, the second difference is 57 - 53 = 4, and the third difference is 53 - 48 = 5. Subsequently, the next difference should be 6, then 7 and finally 8.

Step 3: Calculating the Missing Numbers

Subtract the sequence of differing amounts from the last given number. So the next number would be 48 - 6 = 42, the following number would be 42 - 7 = 35, and the final number would be 35 - 8 = 27

Key Concepts

Pattern Recognition

Understanding pattern recognition in sequences is a crucial mathematical skill. It involves looking at a series of numbers and determining the rule that governs the progression from one number to the next. In the sequence provided (60, 57, 53, 48), recognizing the pattern requires analyzing the differences between consecutive terms.

When we subtract each number from its predecessor, we find that these differences are increasing by one each time. This increasing difference is a common type of pattern in arithmetic sequences. Mastery of pattern recognition can empower students to predict subsequent numbers in a sequence and solve complex problems with ease.

Identifying Sequences

Identifying sequences involves more than just spotting the rule; it requires an understanding of the types of sequences. The exercise given presents an arithmetic sequence because the difference between terms changes consistently—in this case, incrementally by one.

This could lead us to erroneously identify the sequence as linear, with a constant difference, but closer inspection reveals it's not just simply subtracting the same value each time. Recognizing arithmetic sequences is an essential skill, as it sets the foundation for understanding more complex sequence types, such as geometric or Fibonacci sequences.

Sequential Subtraction

Sequential subtraction plays a role in arithmetic sequences, where the difference between consecutive terms is constant or changes by a consistent amount. In our sequence, each term is reduced by an incrementally increasing subtractive value (3, 4, 5, and so on).

Understanding this allows us to continue the pattern logically—calculating the next numbers by subtracting 6, 7, and 8 from the last known term. This process is a straightforward but powerful arithmetic tool, facilitating the solution to problems involving sequential patterns. With practice, sequential subtraction becomes an intuitive step in exploring numerical sequences.