Question

Find the next three terms of each sequence. $$12,8,4,0, \ldots$$

Short answer

The next three terms are -4, -8, and -12.

Step-by-step solution

Identify the Pattern

First, observe the given sequence: 12, 8, 4, 0. Notice the difference between each consecutive pair of terms. The difference is consistent: 12 to 8 is -4, 8 to 4 is -4, and 4 to 0 is -4. Thus, each term decreases by 4.

Extend the Pattern

Using the pattern identified in Step 1, continue the sequence by subtracting 4 from the last known term: 0. So, 0 minus 4 equals -4. The next term is -4.

Determine the Second New Term

Continue using the pattern. Subtract 4 from -4: \(-4 - 4 = -8\). Therefore, the term following -4 is -8.

Determine the Third New Term

Subtract 4 from -8 as per the pattern: \(-8 - 4 = -12\). So, the final term in the required continuation of the sequence is -12.

Key concepts

Recursive Patterns

Recursive patterns provide a step-by-step approach to understanding sequences. This means that each term in the sequence is dependent on the previous one. The rule defines how to move from one term to the next.
When dealing with an arithmetic sequence like the one given, the rule is often a simple addition or subtraction between terms. In this exercise's sequence, the pattern is a subtraction of 4 between terms. By identifying this rule, we can easily determine all future terms.

• To recognize a recursive pattern, find a consistent change between terms.
• This change or rule is applied repeatedly in a sequence.

Recursive patterns help to predict values without calculating each step individually. They are powerful tools for sequences, especially when they follow a consistent structure.

Subtraction Patterns

A subtraction pattern is when each term in the sequence is obtained by subtracting a specific number from the previous term. In our example sequence (12, 8, 4, 0,...), we can see that each number is obtained by subtracting 4 from the previous term.
This type of pattern is a common feature of arithmetic sequences. Recognizing subtraction patterns lets us quickly extend a sequence without recalculating each number.

• Identify the difference between terms, which here is -4.
• Apply this consistent subtraction to continue the sequence.

Subtraction patterns simplify complex series, making it straightforward to find subsequent terms by using the same subtraction rule over and over.

Number Sequences

Number sequences are ordered lists of numbers that often follow a specific rule or pattern. Understanding these sequence rules helps in predicting future numbers in the list. In arithmetic sequences, the difference between consecutive terms is constant. Our sequence shows a common arithmetic progression where terms decrease by 4.
Each number in a sequence is called a "term." By identifying the pattern, we can add or subtract a consistent value to find new terms. This makes it easier to identify how number sequences will continue.

• Determine the constant change between terms, like the -4 in our sequence.
• Utilize this rule to extend the sequence logically.
• Recognizing such patterns is crucial for deciphering number sequences.

Number sequences are indeed all about recognizing and applying consistent patterns, making them a backbone concept in mathematics.