Problem 16 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Determine whether these terms can be the terms of a geometric sequence. If the sequence is geometric, find the common ratio. $$432,-144,48,-16, \dots$$

Step-by-Step Solution

Answer

Yes, it's a geometric sequence with a common ratio of -1/3.

Step 1: Identify the sequence terms

Given sequence terms are: 432, -144, 48, -16.

Step 2: Calculate the ratios

Check if the ratios of consecutive terms are constant by calculating each ratio: - Calculate the ratio of second term to first term: \( r_1 = \frac{-144}{432} = -\frac{1}{3} \).- Calculate the ratio of third term to second term: \( r_2 = \frac{48}{-144} = -\frac{1}{3} \).- Calculate the ratio of fourth term to third term: \( r_3 = \frac{-16}{48} = -\frac{1}{3} \).

Step 3: Determine if the sequence is geometric

Since all computed ratios \( r_1, r_2, \) and \( r_3 \) are equal to \(-\frac{1}{3}\), the sequence is geometric.

Step 4: Identify the common ratio

The common ratio for the geometric sequence is \( r = -\frac{1}{3} \).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key feature. It is the constant factor between any two consecutive terms in the sequence.
Whenever you multiply a term by this ratio, you get the next term in the sequence.
Let's illustrate this. Assume we have a geometric sequence starting with a term \( a_1 \). If the common ratio is \( r \), the following terms would be \( a_2 = a_1 \times r \), \( a_3 = a_2 \times r \), and so on.
This sequence extends further having every term as the product of the previous term and \( r \). In the given exercise:

• The terms provided are: 432, -144, 48, and -16.
• By computing the ratio for each consecutive pair, we found \( r = -\frac{1}{3} \).

So, for this sequence, multiplying each term by \( -\frac{1}{3} \) gives the subsequent term.

Sequence Terms

A sequence in mathematics is simply a list of numbers arranged in a special order.
Each number in the sequence is called a "term". In a geometric sequence, these terms follow a specific rule where each term after the first is produced by multiplying the previous one by a fixed number.
To break it down:

• The first term \( a_1 \) in our example is 432.
• The second term \( a_2 \) is -144, derived by multiplying the first term by the common ratio \( -\frac{1}{3} \).
• This pattern continues for the third term \( 48 \) and the fourth term \( -16 \).

This consistent pattern of multiplication confirms the sequence is geometric.

Ratios of Consecutive Terms

One critical method to confirm if a sequence is geometric is by checking the ratios of consecutive terms.
This involves taking each pair of neighboring terms, dividing the latter by the former, and observing if these ratios are identical.
For the given sequence:

• The ratio \( \frac{-144}{432} = -\frac{1}{3} \) is found between the first two terms.
• The ratio \( \frac{48}{-144} = -\frac{1}{3} \) is found between the second and third terms.
• The ratio \( \frac{-16}{48} = -\frac{1}{3} \) is between the third and fourth terms.

Since all these ratios are equal, this confirms the sequence is geometric. In this case, the repeated ratio \(-\frac{1}{3}\) is what we call the common ratio, showcasing the link between all terms.