Problem 86 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic or geometric, find the next term. (a) \(1,-1,1,-1, \ldots\) (b) \(\sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots\) (c) \(2,-1, \frac{1}{2}, 2, \dots\) (d) \(x-1, x, x+1, x+2, \ldots\)

Step-by-Step Solution

Answer

(a) Geometric, next term 1; (b) Neither; (c) Neither; (d) Arithmetic, next term \(x + 3\).

Step 1: Determine if (a) is Arithmetic or Geometric

The sequence is 1, -1, 1, -1. For an arithmetic sequence, the differences between consecutive terms should be constant. Here, the differences are -2, 2, -2, so it's not constant. For a geometric sequence, each term should be a constant multiple of the previous term. The ratio is -1 (i.e., -1/1, 1/-1), and it is consistent. Therefore, it is a geometric sequence with a common ratio of -1.

Step 2: Find Next Term for (a)

Since the sequence is geometric with a common ratio of -1, multiply the last term by -1 to find the next term. The next term is \(-1 \times (-1) = 1\).

Step 3: Determine if (b) is Arithmetic or Geometric

The sequence is \(\sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1\). Calculate the differences: \(\sqrt[3]{5} - \sqrt{5}, \sqrt[6]{5} - \sqrt[3]{5}, 1 - \sqrt[6]{5}\). These are not constant, so it's not arithmetic.Calculate the ratios: \(\frac{\sqrt[3]{5}}{\sqrt{5}}, \frac{\sqrt[6]{5}}{\sqrt[3]{5}}, \frac{1}{\sqrt[6]{5}}\). These are not constant either, so it's not geometric and thus neither sequence.

Step 4: Determine if (c) is Arithmetic or Geometric

The sequence is 2, -1, \(\frac{1}{2}\), 2. Calculate differences: \(-1 - 2 = -3\), \(\frac{1}{2} + 1 = \frac{3}{2}\), \(2 - \frac{1}{2} = \frac{3}{2}\). Differences are not constant, so not arithmetic. Calculate ratios: \(-\frac{1}{2}\), \(\frac{-1}{2} \div (-1) = \frac{1}{2}\), \(\frac{2}{1} = 2\). Ratios not constant, so neither sequence.

Step 5: Determine if (d) is Arithmetic or Geometric

The sequence is \(x - 1, x, x + 1, x + 2\). Calculate differences: \(x - (x - 1) = 1\), \((x + 1) - x = 1\), \((x + 2) - (x + 1) = 1\). Differences are constant, so it's arithmetic.Since it is arithmetic with a common difference of 1, the sequence is arithmetic.

Step 6: Find Next Term for (d)

For the arithmetic sequence \(x - 1, x, x + 1, x + 2\) with a common difference of 1, the next term after \(x + 2\) is \((x + 2) + 1 = x + 3\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers with a specific pattern. In this pattern, each term is created by adding a constant value, called the "common difference," to the previous term.
For example, in the sequence given in the exercise \(x-1, x, x+1, x+2\), the common difference is 1 because every term is the result of adding 1 to the term preceding it.
The formula to find the next term, if you know the common difference, is to simply add it to the last known term. So, if our last term is \(x+2\), the next term would be \(x+2+1=x+3\).
Arithmetic sequences are straightforward once you identify the pattern. Look for that constant difference in numbers and you'll see the sequence.

Geometric Sequence

A geometric sequence is all about multiplication. Instead of adding like with arithmetic sequences, here you multiply each term by a constant known as the "common ratio."
A classic example is provided in part (a) of the exercise, which is \(1, -1, 1, -1\). You can see that each term is multiplied by \(-1\) to get the next term, making the common ratio \(-1\).
In a geometric sequence, the formula to determine the next term is multiplying the last given term by the common ratio. So, to find the next term of \(-1\), we calculate \(-1 imes -1 = 1\). This repetitive multiplication by a constant reveals the geometric nature of the sequence.

Common Difference

The common difference is a key concept used exclusively in arithmetic sequences. It represents the fixed amount added to each term to get to the next one. This consistency in addition forms the backbone of arithmetic sequences.
As we explored in sequence (d) of the exercise, the common difference is \(1\). Looking further: the transition from \(x-1\) to \(x\) involves adding \(1\), as does moving from \(x\) to \(x+1\).
To double-check that a sequence is arithmetic, check if this "difference" is consistent between all consecutive terms. Keep this constant spot-checking as a helpful method to recognize arithmetic sequences quickly.

Common Ratio

In geometric sequences, the constant multiplier that links each term to the next is called the "common ratio." This consistency is what makes a sequence geometric.
Let's revisit the sequence from part (a), we saw that the common ratio was \(-1\). Simply put, multiply \(1\) by \(-1\) to get \(-1\), then \(-1 \times -1 = 1\), and so on.
It's important to realize that the common ratio can be a fraction too, which often shows up in sequences not covered here. To identify the common ratio, divide any term by the preceding term. Keeping this multiplication rule is the key to understanding geometric sequences.