Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Given a geometric sequence with \(a_{2}=3\) and \(a_{5}=-81\), find \(r\) and \(a_{9}\).

Step-by-Step Solution

Answer

The common ratio \(r\) is \(-3\) and the ninth term \(a_9\) is \(-6561\).

Step 1: Understand the Problem

We are given two terms of a geometric sequence: the second term \(a_2 = 3\) and the fifth term \(a_5 = -81\). We need to find the common ratio \(r\) and the ninth term \(a_9\).

Step 2: Express Terms Using Common Ratio

In a geometric sequence, the \(n\)-th term is given by \(a_n = ar^{n-1}\). Thus, for the second term, we have \(a_2 = ar = 3\), and for the fifth term, \(a_5 = ar^4 = -81\).

Step 3: Divide Equations to Eliminate \(a\)

Dividing the equation for \(a_5\) by the equation for \(a_2\), we get \(\frac{ar^4}{ar} = \frac{-81}{3}\). Simplifying, this yields \(r^3 = -27\).

Step 4: Solve for Common Ratio \(r\)

To solve \(r^3 = -27\), we take the cube root of both sides. Thus, \(r = \sqrt[3]{-27} = -3\).

Step 5: Find the First Term \(a\)

Using \(a_2 = ar = 3\) and \(r = -3\), we find \(a = \frac{3}{-3} = -1\).

Step 6: Calculate the Ninth Term \(a_9\)

Using the formula \(a_n = ar^{n-1}\), we substitute \(n=9\), \(a=-1\), and \(r=-3\) to find \(a_9 = (-1)(-3)^{8} = -1 \times 6561 = -6561\).

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a key element that helps identify the relationship between consecutive terms. It is the constant factor that you multiply by each term to get the next term in the sequence. To find this ratio, divide any term in the sequence by the previous term.
For example, if you have a sequence where every term is obtained by multiplying the previous term by -3, then the common ratio is -3. In our exercise, given \(a_2 = 3\) and \(a_5 = -81\), the common ratio \(r\) is found by dividing the expression for \(a_5\) by \(a_2\), ultimately calculating \(r^3 = -27\), leading to \(r = -3\) after taking the cube root.

Calculating the Terms of a Sequence

Calculating the terms in a geometric sequence involves applying the formula \(a_n = ar^{n-1}\). This formula is central as it relates the term you are looking for (\(a_n\)) to the first term (\(a\)) and the common ratio (\(r\)).
Knowing two specific terms of the sequence allows us to require this formula to first find the common ratio, as seen in the exercise where \(r = -3\). Once \(r\) is determined, any other term such as \(a_9\) can be calculated by substituting \(n = 9\), \(a = -1\), and \(r = -3\) into the formula, resulting in \(-6561\).
This process illustrates the power of the formula for finding any term in the sequence instantly with the known parameters.

Role of the Cube Root in Geometric Sequences

The cube root emerges in geometric sequences when we need to solve for the common ratio in situations where the powers of \(r\) are involved. During the problem, we reached the equation \(r^3 = -27\) through the division method, simplifying the available terms.
Finding the cube root is essential in discovering the common ratio "step by step" when it is raised to a power, as in this case. Solving \(r^3 = -27\) requires the knowledge that the cube root of \(-27\) is \(-3\).
Cube roots become vital when determining \(r\) directly from powers of it, making understanding them important for more complex geometric problems.

Using the Sequence Formula Effectively

The sequence formula \(a_n = ar^{n-1}\) is a comprehensive tool used to calculate any term in a geometric sequence. It hinges on knowing the first term and the common ratio. The sequence formula shows the relationship between any term and the previous terms systematically.
This formula serves not just in determining individual terms but also in deciphering the entire behavior of the sequence. In this exercise, \(a_2 = 3\) helped determine \(r\) when combined with \(a_5 = -81\), and subsequently \(a = -1\), which allowed \(a_9\) to be calculated.
Understanding how to manipulate the sequence formula, using known values, unlocks the full set of possibilities within geometric sequence problems.