Problem 79 - College Algebra with Modeling and Visualization

Question

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$2,8,32,128,512$$

Step-by-Step Solution

Answer

The sequence is geometric.

Step 1: Identify Differences

Identify the differences between consecutive terms in the sequence. The terms given are 2, 8, 32, 128, and 512. The differences are: - 8 - 2 = 6 - 32 - 8 = 24 - 128 - 32 = 96 - 512 - 128 = 384 Since the differences are not constant, the sequence is not arithmetic.

Step 2: Identify Ratios

Identify the ratios between consecutive terms in the sequence. The ratios are: - \( \frac{8}{2} = 4 \)- \( \frac{32}{8} = 4 \)- \( \frac{128}{32} = 4 \)- \( \frac{512}{128} = 4 \)Since the ratios are constant, the sequence is geometric with a common ratio of 4.

Step 3: Conclusion

Since the sequence has a common ratio but not a common difference, it is classified as a geometric sequence.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where each term after the first is generated by adding a constant difference to the previous term. This means that if you subtract one term from the next, you'll always get the same value, known as the 'common difference.' For example, if you have the sequence 3, 6, 9, 12, and the common difference is 3 (each number increases by 3). This continues indefinitely unless you specify a stopping point, or it is a finite sequence.

• Example: Sequence: 5, 10, 15, 20...
• Common Difference: 5
• Formula: If the first term is given as \(a_1\) and the common difference is \(d\), then the \(n\)-th term \(a_n\) can be found using the formula: \(a_n = a_1 + (n-1) \cdot d\)

Identifying an arithmetic sequence is straightforward by checking the difference between any two consecutive terms in the sequence. In the provided example, the sequence was not arithmetic because the differences weren't constant. Understanding this helps in distinguishing between different sequence types easily.

Geometric Sequence

A geometric sequence is a sequence of numbers where each number after the first is found by multiplying the previous one by a fixed, non-zero number called the 'common ratio.' Unlike arithmetic sequences, geometric sequences grow or shrink exponentially based on the value of the ratio. If the common ratio is greater than 1, the sequence will grow larger with each successive term; if it is between 0 and 1, the sequence will decrease.

• Example: Sequence: 2, 4, 8, 16...
• Common Ratio: 2
• Formula: If the first term is \(a_1\) and the common ratio is \(r\), the \(n\)-th term can be found using: \(a_n = a_1 \cdot r^{n-1}\)

To identify a geometric sequence from a group of numbers, you calculate the ratio of consecutive terms. If this ratio remains constant throughout, you have a geometric sequence. In our original exercise, the sequence 2, 8, 32, 128, 512 had a constant ratio of 4, classifying it as a geometric sequence.

Finite Sequence

A finite sequence is simply a sequence that contains a limited number of terms. These sequences can be arithmetic, geometric, or neither. The main characteristic of a finite sequence is that it terminates. Every finite sequence can be explicitly written out since it contains only a certain number of elements, making it easier to analyze.

• Example: Sequence: 1, 2, 3, 4, 5 - This sequence stops at the number 5.
• Identification: To identify a finite sequence, one merely needs to observe its length—finite sequences do not continue indefinitely.

In the exercise provided, the sequence of numbers 2, 8, 32, 128, 512 is finite because it contains only five terms. It's crucial to distinguish between finite sequences and infinite ones, as finite sequences allow for a full exploration of their properties without considering limits or asymptotic behavior.