Problem 40 - Algebra and Trigonometry

Question

The common ratio in a geometric sequence is \(\frac{3}{2},\) and the fifth term is \(1 .\) Find the first three terms.

Step-by-Step Solution

Answer

The first three terms are \( \frac{16}{81}, \frac{8}{27}, \text{ and } \frac{4}{9} \).

Step 1: Understand the Geometric Sequence Formula

A geometric sequence is defined by the formula for the nth term: \( a_n = a_1 \cdot r^{n-1} \), where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio.

Step 2: Apply the Formula for the Fifth Term

Since the fifth term \( a_5 = 1 \), apply the formula: \( a_5 = a_1 \cdot \left( \frac{3}{2} \right)^{4} = 1 \). This equation will help us find \( a_1 \).

Step 3: Simplify the Equation

Calculate \( \left( \frac{3}{2} \right)^4 \), which is equal to \( \frac{81}{16} \). Substitute into the equation: \( a_1 \cdot \frac{81}{16} = 1 \).

Step 4: Solve for the First Term \( a_1 \)

Rearrange the equation \( a_1 \cdot \frac{81}{16} = 1 \) to solve for \( a_1 \) by multiplying both sides by \( \frac{16}{81} \): \( a_1 = \frac{16}{81} \).

Step 5: Calculate the Second Term

Use the geometric sequence formula for the second term: \( a_2 = a_1 \cdot r = \frac{16}{81} \cdot \frac{3}{2} = \frac{24}{81} = \frac{8}{27} \).

Step 6: Calculate the Third Term

Similarly, calculate the third term: \( a_3 = a_2 \cdot r = \frac{8}{27} \cdot \frac{3}{2} = \frac{12}{27} = \frac{4}{9} \).

Key Concepts

Understanding the Common Ratio in Geometric Sequences

In a geometric sequence, one of the primary components is the common ratio, which is the factor that each term is multiplied by to get the next term. To identify the common ratio, you can choose any two consecutive terms in the sequence. Divide the latter term by the former term.
For example, in a sequence if the terms were 2, 6, and 18, the common ratio would be \( \frac{6}{2} = 3 \). This means each term in the sequence is three times the previous term.

• Your sequence can increase or decrease based on whether the common ratio is greater or less than one.
• A common ratio less than one leads to a sequence that converges, getting smaller over time.
• A common ratio greater than one will create a sequence that grows without bound.

Understanding the common ratio aids in predicting future terms of the sequence or reconstructing past terms if some are unknown.

Utilizing the nth Term Formula to Solve Geometric Sequences

The nth term formula for a geometric sequence is pivotal in finding any term if you know the first term and the common ratio. Expressed as \( a_n = a_1 \cdot r^{n-1} \), the formula allows you to plug in what you know to find what you don't. Here, \( a_n \) stands for the term you're seeking in the sequence, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
Let's say you want to find the fifth term given that the first term is 2 and the common ratio is 3:

• Using the formula gives: \( a_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 \).
• Calculating further: \( 3^4 = 81 \), so \( a_5 = 2 \cdot 81 = 162 \).

This showcases how effectively the formula works once the variables are known, preventing the need to manually multiply and track each term in the sequence.

Calculating the Fifth Term in Geometric Sequences

Finding a middle term like the fifth term is simple once you master the use of the nth term formula. Begin by recognizing the relationship between terms: the fifth term, in our given problem, is known to be 1. It's found using the nth term formula, which in this exercise helped to backtrack and discover the first term.
Once you have the first term and the common ratio:

• For example, the formula for the fifth term is \( a_5 = a_1 \cdot r^{4} \).
• Substituting the known values: \( a_1 \cdot \left(\frac{3}{2}\right)^4 = 1 \), and solving gives \( a_1 = \frac{16}{81} \).
• From there, move forward or back again to determine any other terms as needed.

This calculated approach saves time and ensures accuracy over guessing. It allows students to easily navigate through the sequence to elucidate unknown terms.