Problem 39 - Algebra 2

Question

Find the next two terms of each geometric sequence. $$ \frac{1}{3}, \frac{5}{6}, \frac{25}{12}, \dots $$

Step-by-Step Solution

Answer

The next terms are \( \frac{125}{24} \) and \( \frac{625}{48} \).

Step 1: Identify the Pattern in the Sequence

To find the next terms in a geometric sequence, first identify the common ratio between terms. A geometric sequence is defined by the formula \( a_n = a_1 \cdot r^{n-1} \), where \( r \) is the common ratio. Calculate the common ratio by dividing the second term by the first term: \( r = \frac{5/6}{1/3} = \frac{5}{2} \).

Step 2: Verify the Common Ratio

Check that this common ratio applies to the rest of the sequence. Divide the third term by the second term: \( \frac{25/12}{5/6} = \frac{25}{12} \times \frac{6}{5} = \frac{5}{2} \). Both calculations confirm the common ratio is \( \frac{5}{2} \).

Step 3: Calculate the Next Term

Now that we know the common ratio \( r \) is \( \frac{5}{2} \), find the fourth term by multiplying the third term by the common ratio: \( \frac{25}{12} \times \frac{5}{2} = \frac{125}{24} \).

Step 4: Calculate the Fifth Term

To find the fifth term, multiply the fourth term by the common ratio: \( \frac{125}{24} \times \frac{5}{2} = \frac{625}{48} \).

Step 5: Summarize the Next Two Terms

The next two terms in the sequence after \( \frac{1}{3}, \frac{5}{6}, \frac{25}{12} \) are \( \frac{125}{24} \) and \( \frac{625}{48} \).

Key Concepts

Common Ratio

In a geometric sequence, each term is related to the preceding term by a constant factor known as the common ratio. This ratio is critical, as it helps you determine the following terms in the sequence. You can find the common ratio by dividing any term by the previous one.
In the example sequence \[ \frac{1}{3}, \frac{5}{6}, \frac{25}{12}, \ldots \], you find the common ratio \( r \) by performing the division \( \frac{5/6}{1/3} = \frac{5}{2} \). This means each term is \( \frac{5}{2} \) times the previous term. Always remember to verify the common ratio by checking with other terms in the sequence to ensure consistency.
The common ratio plays a fundamental role in defining the sequence and allows for a predictable pattern that can be used to calculate additional terms.

Sequence Patterns

When working with sequences, one of the key tasks is identifying the pattern or rule governing the sequence. In a geometric sequence, this pattern is set by multiplying by the common ratio to move from one term to the next.
The sequence in our example: \[ \frac{1}{3}, \frac{5}{6}, \frac{25}{12}, \ldots \] can be extended by recognizing that each term is multiplied by the common ratio \( \frac{5}{2} \).
Recognizing sequence patterns helps solve real-world problems, such as predicting future events based on current trends or understanding growth processes. It's important to be comfortable identifying and applying these patterns to extend sequences.

Algebraic Expressions

Geometric sequences can often be expressed using algebraic expressions, providing a formulaic way to calculate any term in the sequence. The generic formula for a geometric sequence is \[ a_n = a_1 \cdot r^{n-1} \], where \( a_n \) is the \( n^{th} \) term, \( a_1 \) is the first term, and \( r \) is the common ratio.
Using our sequence, the first term \( a_1 \) is \( \frac{1}{3} \) and the common ratio \( r \) is \( \frac{5}{2} \). If you want to find the fourth term, substitute into the formula:\[ a_4 = \frac{1}{3} \cdot \left( \frac{5}{2} \right)^3 \]. This algebraic approach allows you to calculate any desired term without listing all preceding terms. Understanding these expressions empowers you to work more efficiently with sequences and solve complex problems.