Problem 56 - Prealgebra

Question

Find the next term in each list. $$45,42,39,36,33, \dots$$

Step-by-Step Solution

Answer

The next term is 30.

Step 1: Identify Pattern

Observe the sequence given: 45, 42, 39, 36, 33. The pattern in this sequence is based on subtraction. To find it, subtract each number from the one before it.

Step 2: Subtract Consecutive Terms

Calculate the difference between consecutive terms: \( 45 - 42 = 3 \), \( 42 - 39 = 3 \), \( 39 - 36 = 3 \), \( 36 - 33 = 3 \). The pattern is a consistent decrease of 3.

Step 3: Apply the Pattern to Find Next Term

To find the next term, subtract 3 from the last term provided in the sequence. Start with 33: \( 33 - 3 = 30 \).

Key Concepts

Pattern Recognition

In mathematics, recognizing patterns involves identifying similarities or trends in sets of numbers, shapes, or sequences. Pattern recognition allows us to predict future elements, which is essential in problem-solving and mathematical reasoning.

In the sequence 45, 42, 39, 36, 33, we need to understand how these numbers relate to each other. Recognizing a pattern often requires looking at how each number is transformed to the next. This may involve operations like addition, subtraction, multiplication, or more complex transformations.

Here, **pattern recognition** reveals a sequence where each number is derived by decreasing the previous number. Recognizing that the sequence decreases is the first step to identifying how it progresses. It's critical to observe each transformation to correctly understand the rule governing the sequence.

Subtraction

Subtraction is a fundamental arithmetic operation where one number is taken away from another, resulting in their difference. Understanding subtraction is central to recognizing numerical patterns, especially when dealing with decreasing sequences.

For the sequence 45, 42, 39, 36, 33, subtraction is used to determine the consistent pattern. By calculating the difference between consecutive numbers:

• \( 45 - 42 = 3 \)
• \( 42 - 39 = 3 \)
• \( 39 - 36 = 3 \)
• \( 36 - 33 = 3 \)

We observe that each difference is the same, indicating that subtraction is uniformly applied. This step not only confirms the existence of a pattern but also guides us on how to extend the sequence to future numbers.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. This difference is known as the "common difference." Recognizing that a sequence follows an arithmetic progression allows us to accurately forecast upcoming terms.

In our given sequence (45, 42, 39, 36, 33), the subtraction results in a consistent decline by a fixed number, which is 3 in this case. This consistency categorizes the sequence as an arithmetic progression with a common difference of -3.

To find the next term in an arithmetic progression, simply apply the same common difference to the last known term. From the sequence, beginning at 33, subtract 3 gives us the next term:

• \( 33 - 3 = 30 \)

Understanding arithmetic progression enables us to extend sequences or solve for unknown terms reliably and with ease.