Question

Find the next three terms of each sequence. $$7,8,11,16, \ldots$$

Short answer

Next terms: 23, 32, 43.

Step-by-step solution

Identify the Pattern

First, observe the differences between consecutive terms in the sequence: from 7 to 8 is +1; from 8 to 11 is +3; from 11 to 16 is +5.

Recognize the Sequence Type

Notice that the differences (1, 3, 5) between terms are increasing by 2 each time, indicating that this is a quadratic sequence where the second difference is constant.

Calculate the Next Term

The difference after 16 should be +7 (continuing the pattern of increasing differences by 2). So, the next term is calculated as 16 + 7 = 23.

Calculate the Subsequent Term

The difference following the +7 should be +9. Thus, the next term is 23 + 9 = 32.

Calculate the Final Term

The next difference should be +11. Hence, the following term is calculated as 32 + 11 = 43.

Key concepts

Sequence Patterns

In mathematics, sequences are ordered lists of numbers that follow specific rules to form a pattern. These patterns can help us predict future numbers in the sequence. One popular type involves linear sequences where the difference between terms is constant. However, quadratic sequences are more complex, as the differences between terms themselves change in a regular pattern. This distinction is crucial because it guides us in identifying the mathematical relationships governing the sequence.

To explore sequence patterns, you first identify how each term changes from one to the next. This involves examining not just each number but the space or interval between numbers. These gaps or differences tell us a story about the mathematical rule involved. If these differences shift consistently, they form a second layer of pattern that reveals the type of sequence. This understanding offers critical insights into the nature of sequences we encounter, especially when solving problems related to them.

Term Differences

Term differences in a sequence can tell us a lot about the type of sequence we're dealing with. In the example of the sequence 7, 8, 11, 16, the differences between consecutive terms are:

• 1 (from 7 to 8)
• 3 (from 8 to 11)
• 5 (from 11 to 16)

At first glance, these numbers indicate that the sequence is not linear since the differences are not constant. Instead, these term differences are increasing by 2 each time, suggesting a quadratic sequence.

The concept of term differences is simple yet powerful. It acts as a first clue in the detective work of identifying what kind of sequence we have. For quadratic sequences specifically, the second differences (or how the differences themselves change) will be consistent. Recognizing this can help you predict subsequent terms, as seen when the difference changes from +5 to +7, demonstrating that the sequence is growing in a predictable manner.

Pattern Recognition

Once you've examined both the terms and their differences, recognizing patterns becomes the next step in solving sequence problems. Pattern recognition is a skill that lets you extend a sequence beyond the numbers you initially have.

In quadratic sequences, the constant second difference serves as a pattern that assures us of the sequence's behavior. By identifying that the second differences are constant at +2, one can confidently predict future terms in the sequence. After confirming this pattern, we predict the next terms:

• After 16, the next term difference is +7, leading to 23.
• Following that, the difference is +9, resulting in 32.
• Finally, a difference of +11 adds up to 43.

Recognizing this pattern is pivotal. It transforms a list of numbers into a logical sequence, making it possible to determine not just "what comes next," but "why it comes next." Such comprehension is invaluable in solving both standard homework problems and more advanced mathematical challenges.