Problem 78

Question

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$5,2,-2,-6,-11$$

Step-by-Step Solution

Verified

Answer

The sequence is neither arithmetic nor geometric.

1Step 1: Identify the Sequence Type

First, we need to determine whether the sequence is arithmetic, geometric, or neither by examining the differences or ratios between its terms.

2Step 2: Check if the Sequence is Arithmetic

For a sequence to be arithmetic, each term must differ from the previous term by a constant amount, called the common difference. Calculate the differences: 1. From 5 to 2: difference is 2 - 5 = -3 2. From 2 to -2: difference is -2 - 2 = -4 3. From -2 to -6: difference is -6 - (-2) = -4 4. From -6 to -11: difference is -11 - (-6) = -5 The differences are not all the same (-3, -4, -4, -5), so the sequence is not arithmetic.

3Step 3: Check if the Sequence is Geometric

For a sequence to be geometric, each term must be the previous term multiplied by a constant amount, called the common ratio. Calculate the ratios: 1. From 5 to 2: ratio is 2/5 2. From 2 to -2: ratio is -2/2 = -1 3. From -2 to -6: ratio is -6/(-2) = 3 4. From -6 to -11: ratio is -11/(-6) which is not a simple expression. The ratios are not all the same (2/5, -1, 3, and not clear for the last ratio), hence the sequence is not geometric.

4Step 4: Conclusion

Since the sequence neither has a constant difference nor a constant ratio, it cannot be classified as arithmetic or geometric. Therefore, the sequence is neither.

Key Concepts

Arithmetic SequenceGeometric SequenceFinite SequenceCommon Difference

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the "common difference." If you see the terms changing by the same amount each time, you're likely looking at an arithmetic sequence.

Example of an arithmetic sequence: 3, 6, 9, 12, 15.
Here, the common difference is 3.

To determine if a sequence is arithmetic, you subtract each term from the subsequent one. If these differences are equal, you have an arithmetic sequence. For instance, examining our example sequence 5, 10, 15, the differences are:

10 - 5 = 5
15 - 10 = 5

Because the differences are the same, this sequence is arithmetic with a common difference of 5.

Geometric Sequence

A geometric sequence operates on a different principle compared to an arithmetic sequence. Instead of addition, it involves multiplication. In a geometric sequence, each term is produced by multiplying the previous term by a fixed, non-zero number known as the "common ratio."

Example of a geometric sequence: 2, 4, 8, 16, 32.
Here, the common ratio is 2.

To check if a sequence is geometric, you divide each term by its preceding term. All resulting values should be the same. Consider this sequence: 3, 9, 27, 81. Let's find the ratios:

9 / 3 = 3
27 / 9 = 3
81 / 27 = 3

Since all resulting ratios are 3, this is a geometric sequence with a common ratio of 3.

Finite Sequence

A finite sequence is a sequence that has a limited number of terms. Unlike infinite sequences that go on forever, finite sequences have a start and an end. This means you can count the exact number of elements in a finite sequence.

Consider the sequence 7, 14, 21. This sequence is finite because it clearly begins with 7 and ends with 21. You can enumerate all terms within it.

Finite sequence: 5, 10, 15, 20 (four terms).
Each term is accounted for.

Finite sequences are often easier to deal with because of their limited scope. They are commonly used in arithmetic and geometric sequence problems to explore various properties over a set number of elements.

Common Difference

The common difference is a crucial element in understanding arithmetic sequences. It defines the constant value that is added or subtracted from one term to reach the next. If a sequence consistently adds this fixed value, it is designated as an arithmetic sequence.