Question

Find the next three terms of each sequence. $$9,6,3,0, \ldots$$

Short answer

The next three terms are -3, -6, and -9.

Step-by-step solution

Identify the pattern

Begin by looking at the changes between consecutive terms in the sequence: 9 to 6, 6 to 3, and 3 to 0. The difference between each pair of consecutive terms is 9 - 6 = 3, 6 - 3 = 3, and 3 - 0 = 3. This shows a consistent pattern of subtracting 3 from each term to get the next term.

Apply the pattern

Now, apply the pattern of subtracting 3 to the last known term, which is 0. Subtract 3 from 0 to find the next term in the sequence:\[ 0 - 3 = -3. \]

Continue the pattern

Continue to apply the pattern to find the following terms. Subtract 3 from the latest term (-3):\[ -3 - 3 = -6. \]

Find the final next term

Subtract 3 from the most recent term (-6):\[ -6 - 3 = -9. \]

List the next three terms

The next three terms of the sequence are -3, -6, and -9, based on consistently subtracting 3 from the previous term.

Key concepts

Subtraction Pattern

A subtraction pattern in an arithmetic sequence involves reducing each term by a constant value to find the next term. In simpler terms, each subsequent number in the series is obtained by subtracting a specific number from the one before it.
For example, in the sequence given: 9, 6, 3, 0, each term decreases by 3. You can think of subtraction like this:

• From 9, subtract 3 to reach 6.
• From 6, subtract 3 to reach 3.
• From 3, subtract 3 to reach 0.

This constant difference of 3 dictates the pattern that connects all the terms. It's essential for students to recognize these subtractions, as it helps in predicting the progression of numbers.

Sequence Analysis

Analyzing a sequence is about examining how the numbers change from one term to the next. It often involves looking closely at the difference, sum, or ratio between consecutive terms.
In our exercise, the sequence is 9, 6, 3, 0. The analysis begins by identifying the consistent decrease of 3 between each number. This consistent difference is crucial to understand the arithmetic sequence.
Here are the steps to perform a sequence analysis:

• Compare terms: Look at each pair of consecutive terms and calculate the difference. For instance, check the difference between 9 and 6.
• Determine the common pattern: Once you calculate these differences, notice any repetition.
• Predict future terms based on the established pattern: If consistent, apply it to forecast subsequent numbers.

Through sequence analysis, you gain the insight needed to continue or complete the sequence with confidence.

Mathematical Patterns

Mathematical patterns, like the one in the provided sequence, are foundational elements in learning mathematics. These patterns are arrangements that repeat themselves in a logical sequence.
This particular arithmetic sequence exhibits a pattern of subtraction, where each term is reduced by a fixed number. Recognizing these patterns comes from high levels of observation and practice.
Here’s how to make sense of mathematical patterns:

• Identify uniform changes as seen in our sequence where 3 is always subtracted.
• Apply these patterns to extend the sequence or solve problems.
• Realize the broader application: Understanding these sequences helps in various fields of math and science, enhancing problem-solving skills.

Grasping such patterns sharpens mathematical thinking, making it easier to work through diverse mathematical challenges with an established method.