Problem 61 - Algebra 2

Question

Which geometric sequence DOES NOT include the term 100\(?\) $$ \begin{array}{ll}{\text { A. } 5,10,20, \ldots} & {\text { B. } 337.5,225,150, \ldots} \\ {\text { C. } a_{1}=25, a_{n}=2 a_{n-1}} & {\text { D. } a_{n}=4 \cdot 5^{n}}\end{array} $$

Step-by-Step Solution

Answer

The geometric sequence that does not include the term 100 is sequence B.

Step 1: Analyze sequence A

The sequence is \(5, 10, 20, \ldots\). It is clear that the common ratio is 2, and since \(100 = 5 \cdot 2^n\) for a certain integer \(n\), the number 100 can be a term in this sequence.

Step 2: Analyze sequence B

The sequence is \(337.5, 225, 150, \ldots\). It is clear that the common ratio is \(225/337.5 = 150/225 = 2/3\). Since \(100 \neq 337.5 \cdot (2/3)^n\) for any given \(n\), the number 100 cannot be a term in this sequence. Thus this is the sequence we are looking for.

Step 3: Check other sequences

While we already have our answer, let's quickly check the other sequences just for completeness. For sequence C, \(a_1 = 25\) and \(a_n = 2a_{n-1}\). We can see that \(100 = 2^2 \cdot 25\), a term that could appear in the sequence. As for sequence D, \(a_n = 4 \cdot 5^n\) where 100 can be reached when \(n=1\). Therefore, Sequences A, C, and D all contain 100 as a term.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key feature. It is the factor by which we multiply each term to get to the next one. For sequence A from the original exercise, the common ratio is 2. This is how we go from 5 to 10, and from 10 to 20.
To find the common ratio, you divide any term in the sequence by the previous term. For example, in Sequence B:

• The second term is 225
• The first term is 337.5

When you divide 225 by 337.5, you get a common ratio of 2/3. It helps in predicting future or past terms as long as the ratio stays consistent.
Identifying this ratio is crucial for understanding the sequence's behavior.

Sequence Term

In any sequence, each number is called a term. To identify specific terms in a geometric sequence, we use formulas that incorporate the common ratio and the position of the term within the sequence.

• For instance, in sequence C from the exercise, we know the first term is 25.
• Given that each term is twice the previous, say, if you want to find the fourth term:
• You'd multiply the first term by the common ratio repeatedly:

\( a_4 = 25 \times 2 \times 2 \times 2 = 25 \times 2^3 = 200 \)
This formula \( a_n = a_1 \times r^{n-1} \) where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number, systematically helps identify any term in the sequence.

Exponential Growth

Geometric sequences often demonstrate exponential growth or decay, depending on whether the common ratio is greater than or less than 1.
This exercise includes examples of both. In sequence A, where each term is multiplied by 2, the sequence grows exponentially.
Sequence B, however, displays exponential decay, as each term is multiplied by 2/3, making each subsequent term smaller.

• With exponential growth, a small number can dangerously large over a few terms.
• In contrast, decay will gradually reduce the values, approaching zero.

Understanding these growth and decay patterns is vital in so many contexts, whether predicting population growth or modeling decreases in radioactivity over time.