Problem 44 - College Algebra

Question

For the following exercises, find the number of terms in the given finite geometric sequence. $$ a_{n}=\\{-1,3,-9, \ldots, 2187\\} $$

Step-by-Step Solution

Answer

The sequence has 8 terms.

Step 1: Identify the First Term and Common Ratio

The first task is to identify the first term of the sequence and the common ratio. The first term \( a_1 \) is the first element of the sequence, which is \( -1 \). To find the common ratio \( r \), divide the second term \( 3 \) by the first term \( -1 \):\[r = \frac{3}{-1} = -3\]Thus, the common ratio is \( -3 \).

Step 2: Write the General Formula for the Geometric Sequence

The general formula for the \( n \)-th term of a geometric sequence is given by:\[a_n = a_1 \cdot r^{n-1}\]Substituting the values for \( a_1 \) and \( r \):\[a_n = -1 \cdot (-3)^{n-1}\]

Step 3: Set the Formula Equal to the Last Term

We are given that the last term of the sequence is \( 2187 \). Therefore, we set the general formula equal to \( 2187 \):\[-1 \cdot (-3)^{n-1} = 2187\]This simplifies to:\[(-3)^{n-1} = -2187\]

Step 4: Solve for \( n-1 \)

Recognize that \( -2187 \) can be expressed as \( (-3)^7 \) because:\[(-3)^7 = -2187\]Hence, we equate and solve for \( n-1 \):\[n-1 = 7\]

Step 5: Solve for \( n \)

Add 1 to find \( n \):\[n = 7 + 1 = 8\]Thus, there are 8 terms in the sequence.

Key Concepts

Finite Sequence

A finite sequence is a list of numbers that has a clear beginning and an end. Let's take a closer look at what makes a sequence finite:

• Starts with a specific first term.
• Ends at a certain term after a definite count of elements.

In our example, the sequence is \(-1, 3, -9,..., 2187\). You can see there's a starting term, \(-1\), and an ending term, \(2187\). Finite sequences have a specific number of terms, which is part of what we are trying to find in problems like these.
A finite sequence can be easy to interpret because you know exactly where it begins and ends. Understanding this aspect is as simple as counting the distinct numbers from start to finish.

Common Ratio

In a geometric sequence, the common ratio is a key detail. It describes how to get from one term to the next by multiplying by a constant value called the common ratio, \(r\). For example:

• If the first term is \(-1\), and the next is \(3\), you determine \(r\) by dividing: \(\frac{3}{-1} = -3\).

This means each term is \(-3\) times the term before it. The concept of the common ratio is simple yet crucial because it defines the pattern of the sequence.
Knowing the common ratio allows you to predict any term in the sequence just by knowing its position in the sequence. This is what makes geometric sequences predictable and easy to work with.

General Formula for N-th Term

The general formula provides a way to calculate any term in a geometric sequence without writing out all the terms. It is given by:

• \(a_n = a_1 \cdot r^{n-1}\)

Here, \(a_1\) is the first term and \(r\) is the common ratio. For example, our sequence's formula becomes \(a_n = -1 \cdot (-3)^{n-1}\) after substituting \(a_1\) as \(-1\) and \(r\) as \(-3\).
This formula lets us find any term directly without listing them all, which is especially useful in sequences with a large number of terms. It simplifies the process significantly, turning a potentially cumbersome task into a straightforward calculation.

Solving for Terms

Finding the number of terms in a finite geometric sequence involves using the general formula. For the sequence ranging from \(-1\) to \(2187\), we start by setting up the equation:

• \(-1 \cdot (-3)^{n-1} = 2187\)

Reorganize this equation into a simpler form:

• \((-3)^{n-1} = -2187\)

Next, recognize that \(-2187\) can indeed be rewritten as \((-3)^7\). This tells us:

• \(n-1 = 7\)

Add \(1\) to both sides to find \(n\), which gives \(n = 8\).
This means the sequence contains 8 terms in total. Solving for terms is about understanding patterns and applying formulas logically. It showcases how math and reasoning work hand-in-hand to uncover the properties of a sequence.