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Chapter 11: Problem 33

Identify each sequence as arithmetic, geometric, or neither. $$1,4,9,16,25, \dots$$

Short Answer

Expert verified
The sequence is neither arithmetic nor geometric.

Step by step solution

01

Identify the Pattern

First, observe the given sequence: 1, 4, 9, 16, 25, Notice that each term is related to the previous term in a certain way.
02

Check for an Arithmetic Sequence

In an arithmetic sequence, the difference between consecutive terms is constant. Calculate the differences between the terms: 4 - 1 = 3 9 - 4 = 5 16 - 9 = 7 25 - 16 = 9 Since the differences (3, 5, 7, 9) are not constant, the sequence is not arithmetic.
03

Check for a Geometric Sequence

In a geometric sequence, the ratio between consecutive terms is constant. Calculate the ratios between the terms: 4 / 1 = 4 9 / 4 = 2.25 16 / 9 鈮� 1.78 25 / 16 = 1.5625 Since the ratios (4, 2.25, 1.78, 1.5625) are not constant, the sequence is not geometric.
04

Conclusion

Since the sequence is neither arithmetic nor geometric, it does not fall into either of these categories.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Sequence
An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is called the 'common difference'. To identify an arithmetic sequence, follow these steps:
  • Calculate the difference between the first and second terms.
  • Repeat this calculation for the following pairs of terms.
  • If the difference is the same for all pairs, then the sequence is arithmetic.
For example, consider the sequence: 5, 8, 11, 14. The differences are:
8 - 5 = 3, 11 - 8 = 3, and 14 - 11 = 3. Since the difference is constant (3), this is an arithmetic sequence.
Geometric Sequence
A geometric sequence is a series of numbers in which the ratio between consecutive terms is constant. This ratio is known as the 'common ratio'. To identify a geometric sequence, follow these steps:
  • Calculate the ratio between the first and second terms.
  • Repeat this calculation for the following pairs of terms.
  • If the ratio is the same for all pairs, then the sequence is geometric.
For example, consider the sequence: 2, 6, 18, 54. The ratios are:
6 / 2 = 3, 18 / 6 = 3, and 54 / 18 = 3. Since the ratio is constant (3), this is a geometric sequence.
Pattern Recognition
Pattern recognition involves identifying the rule that describes the relationship between the terms in a sequence. This can help in categorizing sequences and predicting future terms. For example, in the sequence 1, 4, 9, 16, 25, you can recognize that each term is a perfect square:
  • 1 = 1虏
  • 4 = 2虏
  • 9 = 3虏
  • 16 = 4虏
  • 25 = 5虏
Recognizing this pattern helps us understand the sequence better and identify that it does not fit into the arithmetic or geometric categories but follows a different rule.

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