Question

Describe the pattern in each sequence, and use the pattern to find the next three terms. $$1,2,4,7,11,16, \ldots$$

Short answer

The next three terms are 22, 29, and 37.

Step-by-step solution

Identify the differences between consecutive terms

Start by finding the differences between each consecutive term in the sequence. The sequence provided is 1, 2, 4, 7, 11, 16. Calculate each difference: 2 - 1 = 14 - 2 = 27 - 4 = 311 - 7 = 416 - 11 = 5The differences are 1, 2, 3, 4, 5.

Identify the pattern in the differences

Examine the differences obtained in the previous step. The differences form an arithmetic sequence: 1, 2, 3, 4, 5. Notice that each difference increases by 1.

Predict the next differences

Based on the pattern of differences, predict the next three differences. Since each difference increases by 1, the next three differences will be: 6, 7, 8.

Calculate the next three terms

Add the predicted differences to the last term of the sequence to find the next three terms. The last given term is 16.16 + 6 = 2222 + 7 = 2929 + 8 = 37Therefore, the next three terms in the sequence are 22, 29, and 37.

Key concepts

Pattern Recognition

Pattern recognition is the ability to identify and understand repeating elements within a sequence. It's like seeing the same building blocks being used in different ways. In arithmetic sequences, patterns can often be spotted by observing the relationship between the terms. For instance, if you have a sequence like 1, 2, 4, 7, 11, 16, you can spot a pattern by looking at how each term changes to the next.
Start by calculating the differences between consecutive terms:

• 2 - 1 = 1
• 4 - 2 = 2
• 7 - 4 = 3
• 11 - 7 = 4
• 16 - 11 = 5

These calculated values (1, 2, 3, 4, 5) reveal a pattern where each difference increases by 1. Recognizing such patterns is crucial to understanding how sequences are constructed.

Consecutive Terms

Consecutive terms are simply terms that follow one another in a sequence. Understanding how these terms are connected can help you identify patterns. In the sequence 1, 2, 4, 7, 11, 16, each number (term) follows directly from the previous one. By figuring out how these consecutive terms relate, you can identify the pattern forming between them.
For example, when calculating the difference between each consecutive term:

• 2 follows 1
• 4 follows 2
• 7 follows 4
• 11 follows 7
• 16 follows 11

By breaking down the sequence into simpler consecutive parts, you can more easily recognize the differences and thereby the overall pattern.

Differences in Sequences

The differences between terms in a sequence can reveal a lot about its structure. Once you recognize these differences, you can better understand the nature of the sequence. In our example sequence (1, 2, 4, 7, 11, and 16), the differences are:

• 2 - 1 = 1
• 4 - 2 = 2
• 7 - 4 = 3
• 11 - 7 = 4
• 16 - 11 = 5

These differences form their own sequence of 1, 2, 3, 4, 5 which is an arithmetic sequence where each term increases by 1. Understanding these differences is key to discerning the actual pattern and predicting future terms.

Future Terms Prediction

Predicting future terms in a sequence involves extending the identified pattern. Once you have the differences between terms, you can use these to predict future differences. For the sequence 1, 2, 4, 7, 11, 16, we recognize the differences as 1, 2, 3, 4, 5 and see that each difference increases by 1.
To predict the next differences:

• 5 + 1 = 6
• 6 + 1 = 7
• 7 + 1 = 8

Then, by adding these new differences to the last term (16), we can find the next terms:

• 16 + 6 = 22
• 22 + 7 = 29
• 29 + 8 = 37

Therefore, the next three terms are 22, 29, and 37. Predicting future terms by following the pattern helps extend the sequence accurately.