Problem 86 - Algebra and Trigonometry

Question

Arithmetic or Geometric? 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. $$ \begin{array}{ll}{\text { (a) } 5,-3,5,-3, \ldots} & {\text { (b) } \frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots} \\ {\text { (c) } \sqrt{3}, 3,3 \sqrt{3}, 9,9, \ldots} & {\text { (d) } 1,-1,1,-1, \ldots} \\ {\text { (e) } 2,-1, \frac{1}{2}, 2, \ldots} & {\text { (f) } x-1, x+1, x+2, \ldots} \\\ {\text { (g) }-3,-\frac{3}{2}, 0, \frac{3}{2}, \ldots} & {\text { (h) } \sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots}\end{array} $$

Step-by-Step Solution

Answer

(b) 3, (c) 9\sqrt{3}, (d) 1, (g) 3

Step 1: Checking Arithmetic Sequence for (a)

For the sequence \(5, -3, 5, -3, \ldots\), calculate the difference between consecutive terms: \(-3 - 5 = -8\) and \(5 - (-3) = 8\). Since the differences are not constant, this sequence is not arithmetic.

Step 2: Checking Geometric Sequence for (a)

For the same sequence, calculate the ratio of consecutive terms: \(\frac{-3}{5} = -\frac{3}{5}\) and \(\frac{5}{-3} = -\frac{5}{3}\). The ratios are not constant, so this sequence is not geometric.

Step 3: Determine Next Term for (a)

Since the sequence is neither arithmetic nor geometric, the next term cannot be determined in a systematic way.

Step 4: Checking Arithmetic Sequence for (b)

For the sequence \(\frac{1}{3}, 1, \frac{5}{3}, \frac{7}{3}, \ldots\), calculate the difference: \(1 - \frac{1}{3} = \frac{2}{3}\), \(\frac{5}{3} - 1 = \frac{2}{3}\), \(\frac{7}{3} - \frac{5}{3} = \frac{2}{3}\). The constant difference marks it as arithmetic.

Step 5: Determine Next Term for (b)

Since it is arithmetic with a common difference of \(\frac{2}{3}\), add this to the last term: \(\frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3\). Next term is 3.

Step 6: Checking Arithmetic Sequence for (c)

For the sequence \(\sqrt{3}, 3, 3\sqrt{3}, 9, \ldots\), calculate the differences, which are not constant, disqualifying it as arithmetic.

Step 7: Checking Geometric Sequence for (c)

Calculate the ratio: \(\frac{3}{\sqrt{3}} = \sqrt{3}\), \(\frac{3\sqrt{3}}{3} = \sqrt{3}\), \(\frac{9}{3\sqrt{3}} = \sqrt{3}\). With constant ratio \(\sqrt{3}\), it is geometric.

Step 8: Determine Next Term for (c)

Multiply the last term by the common ratio: \(9 \times \sqrt{3} = 9\sqrt{3}\). The next term is \(9\sqrt{3}\).

Step 9: Checking Arithmetic Sequence for (d)

For the sequence \(1, -1, 1, -1, \ldots\), calculate the difference: \(-1 - 1 = -2\), \(1 + 1 = 2\). The differences alternate, so it is not arithmetic.

Step 10: Checking Geometric Sequence for (d)

Calculate the ratio: \(-1\) and \(-1\). The ratio is constant (-1), marking it as geometric.

Step 11: Determine Next Term for (d)

Multiply the last term by the common ratio: \(-1 \times -1 = 1\). The next term is 1.

Step 12: Checking Arithmetic Sequence for (e)

Calculate differences for the sequence \(2, -1, \frac{1}{2}, 2, \ldots\), finding they are not constant, disqualifying it as arithmetic.

Step 13: Checking Geometric Sequence for (e)

Calculate the ratios: \(-\frac{1}{2}\), \(-\frac{1}{3}\), \(4\). Ratios are not constant, so it is neither.

Step 14: Determine Next Term for (e)

No next term, as the sequence is neither arithmetic nor geometric.

Step 15: Checking for Sequence Pattern (f)

For generic sequence \(x-1, x+1, x+2, \ldots\), compute differences: 2 and 1. Not arithmetic, and without fixed ratio, not geometric.

Step 16: Determine Next Term for (f)

Cannot determine due to lack of pattern establishing rules.

Step 17: Check Arithmetic Sequence for (g)

Calculate differences for \(-3, -\frac{3}{2}, 0, \frac{3}{2}, \ldots\): \(\frac{3}{2}\), \(\frac{3}{2}\), \(\frac{3}{2}\), confirming arithmetic.

Step 18: Determine Next Term for (g)

Add common difference to the last term: \(\frac{3}{2} + \frac{3}{2} = 3\). Next term is 3.

Step 19: Checking Arithmetic Sequence for (h)

For \(\sqrt{5}, \sqrt[3]{5}, \sqrt[6]{5}, 1, \ldots\), differences not constant, ruling out arithmetic.

Step 20: Checking Geometric Sequence for (h)

Calculate ratios, which are not constant. Thus, this sequence is neither arithmetic nor geometric.

Step 21: Determine Next Term for (h)

As it is neither, no next term can be determined systematically.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between each number, known as the common difference. Let's break it down: imagine you have a starting number, and each subsequent number in the sequence is formed by adding the same exact value to the previous number. This "same exact value" is what we call the common difference.

For example, consider the sequence \(5, 8, 11, 14, \ldots\). Here, the difference between each consecutive pair of numbers is always \(3\). That means, to find the next term, you just add \(3\) to the last number in the sequence. Very straightforward!

Arithmetic sequences pop up frequently in everyday life, from calculating the total sum of deposits into a savings account to planning equal steps to complete a task. A real-world intuitive understanding simplifies mastering these sequences.

Geometric Sequences

On the other hand, geometric sequences use multiplication by a fixed number, known as the common ratio, to create each new term in the sequence. Understanding this differs from arithmetic sequences – it's not about what you add, but what you multiply.

For instance, take the sequence \(2, 4, 8, 16, \ldots\). Notice anything? Each term can be identified by multiplying the previous term by \(2\), the common ratio in this sequence. So, if you want to find the next term, multiply \(16\) by \(2\) to get \(32\). Simple once you spot the pattern!

Geometric sequences are vital for concepts such as exponential growth in populations or compound interest in finance – everywhere things grow (or shrink) in scale proportionately.

Common Difference

The common difference in an arithmetic sequence is arguably its most defining feature. It is the consistent number that you add to each term to get the next one. Think of it as the 'step size' between the numbers.

Consider the sequence \(10, 7, 4, 1, \ldots\). Here, if you look closely, every number is obtained by subtracting \(3\). So, in this case, the common difference is \(-3\). The sign of the difference indicates whether the sequence is increasing or decreasing.

The common difference is fundamental in arithmetic progressions, enabling us to predict any number in the sequence with ease, or even find the sum of a whole sequence!

Common Ratio

The equivalent term for geometric sequences is the common ratio. It's what you multiply a term by to arrive at the next term. The common ratio defines the strength or the 'multiplying factor' of the geometric sequence.

For example, in the sequence \(5, 15, 45, 135, \ldots\), each term results from multiplying the previous one by \(3\). So, in this sequence, the common ratio is \(3\).

Recognizing and understanding the common ratio facilitates identifying patterns in data that involve multiplicative processes, such as interest rates or biological cell growth.

Pattern Recognition

Identifying whether a sequence is arithmetic or geometric hinges largely on recognizing patterns. This involves analyzing the changes between the terms and deducing if they represent a constant additive difference or a constant multiplicative ratio.

By systematically comparing differences or ratios, you can unlock the pattern behind a sequence. For an arithmetic pattern, strive to find a common difference. For a geometric one, determine a consistent ratio.

This skill is essential not only in mathematics but in everyday life and science, helping to predict future values, identify trends, and solve complex problems involving sequences.