Problem 37 - Algebra 2

Question

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.) \(19,683 ; \quad ; \quad ; \quad ; 243 ; \ldots\)

Step-by-Step Solution

Answer

The missing terms of the geometric sequence are 728, 27, and 1.

Step 1: Find the common ratio

Let's denote the common ratio by r. We can compute it using the formula for the geometric mean: \(\sqrt{a\cdot b} = m\), where m is the geometric mean (the third term in this case). Substituting the given terms, we have: \(\sqrt{19,683 \cdot 243} = m\). After simplifying, we find that \(m = 729\). In a geometric series, the ratio between any two consecutive terms is constant; hence \(729/19,683 = r\). From these computations, we find that \(r = 0.037\).

Step 2: Use the common ratio to find the missing terms

Knowing the common ratio, we can find the missing second term (being the first term multiplied by the common ratio) as \(19,683 \cdot 0.037 = 728\). Using the same logic, the third term (being the second term multiplied by the common ratio) is \(728 \cdot 0.037 = 27\), and the fourth term (being the third term multiplied by the common ratio) is \(27 \cdot 0.037 = 1\). Hence, we have the missing terms of the geometric sequence.

Key Concepts

common ratio

The common ratio is an essential element in any geometric sequence. In a geometric sequence, each term after the first is derived by multiplying the previous term by a constant called the common ratio. It tells us how much each term scales the previous one.

To find the common ratio, divide any term by the previous term in the sequence. For example, if you have the sequence elements 5, 10, and 20, the common ratio would be \[ \frac{10}{5} = 2 \] or \[ \frac{20}{10} = 2.\]

• The common ratio, denoted as \( r \), remains the same throughout the sequence.
• This ratio can be positive or negative, which can affect the pattern of the sequence terms significantly.
• In the step-by-step solution, the common ratio \( r \) was found by comparing terms, specifically for understanding changes over successive terms.

Finding the common ratio is the key to unraveling the structure of the sequence and helps predict future terms.

geometric mean

The geometric mean is a special type of average, different from the arithmetic mean most people are familiar with. In a geometric sequence, the geometric mean provides us with valuable insight and is typically found between two non-consecutive terms.

To calculate the geometric mean, take the square root of the product of two terms you want between which you want the mean. If \( a \) and \( b \) are the terms, the geometric mean \( m \) is given by:\[ m = \sqrt{a \cdot b}. \]

• The geometric mean focuses more on the multiplicative relationships rather than additive ones.
• In the exercise solution, the third term is considered the geometric mean of the first (19,683) and fifth term (243).
• This method is particularly useful when working with datasets related by proportions.

Understanding how to calculate and use the geometric mean can significantly enhance your ability to navigate and solve problems in sequences.

sequence terms

Sequence terms are simply the individual elements or numbers that make up a sequence. In a geometric sequence, these terms are interconnected through the common ratio, which explains how each subsequent term is derived from its predecessor.

With the provided example, unraveling the unknown terms involved using the common ratio. Given the first term and common ratio, each term of the sequence is calculated by multiplying the preceding term by this ratio.

• Start with an initial known term of the sequence, then continuously multiply by the common ratio.
• Missing terms in a sequence can be filled in by applying the common ratio sequentially from the known terms, as shown with 19,683, 729, 27, and 1.
• This technique allows us to understand the progression of the sequence and predict future terms easily.

Recognizing the structure and pattern between sequence terms is crucial in solving and understanding sequences, especially geometric ones.