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. Find the next term if the sequence is arithmetic or geometric. (a) \(5,-3,5,-3, \dots\) (b) \(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \dots\) (c) \(\sqrt{3}, 3,3 \sqrt{3}, 9, \ldots\) (d) \(1,-1,1,-1, \ldots\) (e) \(2,-1, \frac{1}{2}, 2, \dots\) (f) \(x-1, x, x+1, x+2, \ldots\) (g) \(-3,-\frac{3}{2}, 0, \frac{3}{2}, \dots\) (h) \(\sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots\)

Step-by-Step Solution

Answer

(a) Neither; (b) Arithmetic, next 3; (c) Geometric, next 9√3; (d) Geometric, next 1; (e) Neither; (f) Arithmetic, next x+3; (g) Arithmetic, next 3; (h) Neither.

Step 1: Analyze sequence (a)

The terms given are \(5, -3, 5, -3, \ldots\). To check if it's an arithmetic sequence, subtract consecutive terms: \(-3 - 5 = -8\) and \(5 - (-3) = 8\). The differences aren't constant, so it's not arithmetic. For geometric, divide consecutive terms: \(-3/5\) and \(5/(-3)\). The ratios aren't constant, so it's not geometric either.

Step 2: Analyze sequence (b)

The terms are \(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots\). Subtract consecutive terms: \(1 - \frac{1}{3} = \frac{2}{3}\), \(\frac{5}{3} - 1 = \frac{2}{3}\), and \(\frac{7}{3} - \frac{5}{3} = \frac{2}{3}\). The difference is constant, so it's arithmetic. The common difference is \(\frac{2}{3}\), so the next term is \(\frac{7}{3} + \frac{2}{3} = 3\).

Step 3: Analyze sequence (c)

Terms: \(\sqrt{3}, 3, 3 \sqrt{3}, 9, \ldots\). Dividing terms: \(3/\sqrt{3} = \sqrt{3}\), \((3\sqrt{3})/3 = \sqrt{3}\), and \(9/(3\sqrt{3}) = \sqrt{3}\). The ratio is constant, hence geometric with a ratio \(\sqrt{3}\). Next term is \(9 \times \sqrt{3} = 9\sqrt{3}\).

Step 4: Analyze sequence (d)

The sequence is \(1, -1, 1, -1, \ldots\). Subtract terms: \(-1 - 1 = -2\), \(1 - (-1) = 2\). Different differences, not arithmetic. For geometric, divide: \(-1/1 = -1\), \(1/-1 = -1\). Constant ratio, it's geometric. Next term: \(-1 \times -1 = 1\).

Step 5: Analyze sequence (e)

Terms: \(2, -1, \frac{1}{2}, 2, \ldots\). Subtract terms: \(-1 - 2 = -3\), \(\frac{1}{2} - (-1) = \frac{3}{2}\). Differences aren't constant, not arithmetic. Divide terms: \(-1/2 = -\frac{1}{2}\), \((\frac{1}{2})/(-1) = -\frac{1}{2}\). Changing ratio, hence neither arithmetic nor geometric.

Step 6: Analyze sequence (f)

Terms: \(x-1, x, x+1, x+2, \ldots\). Subtract terms: \(x - (x-1) = 1\), \((x+1) - x = 1\). Constant difference 1, so it's arithmetic. Next term is \(x+2 + 1 = x+3\).

Step 7: Analyze sequence (g)

Sequence: \(-3, -\frac{3}{2}, 0, \frac{3}{2}, \ldots\). Subtract terms: \(-\frac{3}{2} + 3 = \frac{3}{2}\), \(0 + \frac{3}{2} = \frac{3}{2}\). Constant difference, arithmetic, with difference \(\frac{3}{2}\). Next term is \(\frac{3}{2} + \frac{3}{2} = 3\).

Step 8: Analyze sequence (h)

Terms: \(\sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots\). Differences: \(\sqrt[3]{5} - \sqrt{5}\), \(\sqrt[6]{5} - \sqrt[3]{5}\), they're not constant, so not arithmetic. Dividing gives a changing ratio, not geometric, hence it's neither.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is formed by adding a fixed quantity known as the 'common difference' to the previous term. For example, in the sequence 2, 4, 6, 8, ..., each term is created by adding 2 to the previous term. This consistent addition gives the sequence its linear nature, making it easy to predict subsequent terms.
Understanding arithmetic sequences helps in grasping various mathematical concepts and is commonly used in real-world scenarios like calculating total savings with regular deposits or predicting the next terms in linear patterns.
Some key points to remember about arithmetic sequences include:

• The common difference can be positive, negative, or zero.
• If the common difference is zero, every term in the sequence is the same.
• Formally, an arithmetic sequence can be expressed as \(a_n = a_1 + (n-1)d\) where \(a_1\) is the first term and \(d\) is the common difference.

Geometric Sequence

Geometric sequences consist of numbers where each term after the first is obtained by multiplying the previous one by a fixed non-zero quantity called the 'common ratio'. Let's consider the sequence 3, 6, 12, 24, ...; here, each term is multiplied by 2, which represents the common ratio. This multiplication factor differentiates geometric sequences from arithmetic ones, as they grow (or shrink) at an exponential rate.
Geometric sequences are crucial in various areas of mathematics, especially in modeling situations involving exponential growth or decay, such as population increases or radioactive decay.
Here are essential characteristics of geometric sequences:

• The common ratio can be positive or negative, affecting the pattern and alternation of the sequence.
• If the common ratio is one, all the terms of the sequence remain equal.
• Geometric sequences are described by \(a_n = a_1 \, r^{(n-1)}\) where \(a_1\) is the first term, and \(r\) is the common ratio.

Common Difference

The common difference is the key factor in arithmetic sequences, representing the numeric value added to each term to obtain the subsequent one. When examining whether a sequence is arithmetic, it's crucial to check if the subtraction of consecutive terms yields the same result consistently.
For example, consider the sequence 5, 8, 11, 14, ..., where the common difference is 3. This steady difference signifies that the sequence grows linearly, allowing prediction of future terms. Understanding the concept of the common difference can simplify solving problems involving linear patterns.
It's vital to remember:

• Determining the common difference involves checking consecutive term differences.
• If differences vary, the sequence isn't arithmetic.

Common Ratio

A common ratio is a foundational element in defining geometric sequences, providing the multiplier needed to derive one term from another. To identify whether a sequence is geometric, compute the ratio between consecutive terms and check if it remains constant throughout the sequence.
Consider the sequence 2, 4, 8, 16, ..., where dividing any term by its predecessor yields 2, the common ratio. This constant multiplier explains the sequence's rapid growth, a hallmark of exponential change.
Important aspects of the common ratio include:

• A constant common ratio confirms a geometric sequence.
• If the ratio isn't uniform, the sequence may not be geometric.