Problem 78 - Precalculus: Mathematical 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

(b) is arithmetic: next term \(3\); (c) is geometric: next term \(9\sqrt{3}\); (f) is arithmetic: next term \(x+3\); (g) is arithmetic: next term \(3\).

Step 1: Determine if each sequence is arithmetic

An arithmetic sequence has a constant difference between successive terms. For example, in the sequence from part (a) which is \(5, -3, 5, -3\), the differences are \(-3 - 5 = -8\) and \(5 - (-3) = 8\). These are not constant, so it is not arithmetic. Repeat this check for all sequences.

Step 2: Determine if each sequence is geometric

A geometric sequence has a constant ratio between successive terms. For example, for sequence (a) where each term alternates, the ratio is \(-\frac{3}{5}\) and \(-\frac{5}{3}\) between terms, which is not constant. Repeat this calculation for the remaining sequences.

Step 3: Check each sequence individually

For sequences where neither step 1 nor 2 concludes the pattern, manually calculate the relationships between successive terms, and see if any recognizable pattern emerges, such as alternating signs or non-linear progressions.

Step 4: Find the next term if sequence is arithmetic or geometric

For sequences identified as arithmetic, add the common difference to the last known term to find the next term. For geometric sequences, multiply the last term by the common ratio. Example: in sequence (f) which is arithmetic with a common difference of 1, the next term is \(x+2 +1 = x+3\).

Step 5: Conclusion: Analyze each sequence results

Summarize findings for each sequence:(a) Neither arithmetic nor geometric.(b) Arithmetic with a common difference of \(\frac{2}{3}\); next term is \(\frac{3}{3} + \frac{2}{3} = 3\).(c) Geometric with common ratio of \(\sqrt{3}\); next term is \(3\sqrt{3} \times \sqrt{3} = 9\times \sqrt{3}\).(d) Neither arithmetic nor geometric.(e) Neither arithmetic nor geometric due to alternating pattern.(f) Arithmetic with common difference 1; next term is \(x+3\).(g) Arithmetic with common difference \(\frac{3}{2}\); next term is \(3\) since \(\frac{3}{2}+\frac{3}{2}=3\).(h) Neither arithmetic nor geometric due to root pattern complexity.

Key Concepts

Arithmetic Sequence

In an arithmetic sequence, each term is derived from the previous one by adding a fixed number, known as the "common difference." This is a straightforward sequence type with a consistent pattern. Think of it like climbing a ladder – each step is spaced evenly apart.

To determine if a sequence is arithmetic, check if the difference between each successive term remains constant. For example, consider the sequence:

• (b) \(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3} \)

The differences between terms are as follows:

• \(1 - \frac{1}{3} = \frac{2}{3} \)
• \(\frac{5}{3} - 1 = \frac{2}{3} \)
• \(\frac{7}{3} - \frac{5}{3} = \frac{2}{3} \)

Since the common difference is \(\frac{2}{3}\), this confirms it is arithmetic.

If uncertain, look for equal spacing. It's like finding a rhythm – once you recognize it, predicting the next note in the sequence becomes simple.To find the next term, add the common difference to the last term: \[ \frac{7}{3} + \frac{2}{3} = 3 \].

Geometric Sequence

A geometric sequence takes a different approach. Instead of adding, you * multiply* by a fixed number called the "common ratio." Imagine a snowball rolling down a hill, gathering more snow each time it turns, without needing any additional snowflakes added manually. This reflects how each term in a geometric sequence is derived from multiplying the previous term.

To identify a geometric sequence, calculate the ratio of successive terms. For example, take:

• (c) \( \sqrt{3}, 3, 3 \sqrt{3}, 9 \)

The ratios are:

• \( \frac{3}{\sqrt{3}} = \sqrt{3} \)
• \( \frac{3 \sqrt{3}}{3} = \sqrt{3} \)
• \( \frac{9}{3 \sqrt{3}} = \sqrt{3} \)

Since the ratio is consistently \(\sqrt{3}\), this sequence is geometric.

To find the next term, multiply the last term by this common ratio: \[ 9 \times \sqrt{3} = 9\sqrt{3} \]. Notice the pattern growing exponentially, echoing the snowball's unavoidable build-up.

Mathematical Progression

Mathematical progressions are broader sets of sequences, encompassing both arithmetic and geometric. But what if a sequence doesn’t fit neatly into either category? That's where recognizing the distinction between a general progression and its "fixed pattern" siblings is important.

For instance, look at sequence (h):

• \(\sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1 \)

Trying to force a uniform progression, whether by difference or ratio, proves difficult, and this sequence is neither purely arithmetic nor geometric. Such sequences are often the result of combining different mathematical operations. They might appear irregular at first glance, but they can still hold systematic beauty when comprehended fully.

Understanding a mathematical progression requires analyzing each aspect of change in detail. Focus on each transformation – whether through roots, exponents, or more complex patterns. Some sequences like (h) contain layers of mathematical operations that challenge straight categorization, requiring further exploration and a keen eye for uncommon patterns.