Problem 45 - College Algebra

Question

For the following exercises, find the number of terms in the given finite geometric sequence. $$ a_{n}=\left\\{2,1, \frac{1}{2}, \ldots, \frac{1}{1024}\right\\} $$

Step-by-Step Solution

Answer

The sequence has 12 terms.

Step 1: Identify the First Term

Look at the given sequence: \( 2, 1, \frac{1}{2}, \ldots, \frac{1}{1024} \). The first term of the sequence is \( a_1 = 2 \).

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term. Here, \( r = \frac{1}{2} \div 2 = \frac{1}{2} \times \frac{1}{2} = \frac{1}{2} \).

Step 3: Express the General Term

Use the formula for the \( n \)-th term of a geometric sequence: \( a_n = a_1 \cdot r^{n-1} \). For this sequence, it becomes \( a_n = 2 \cdot \left( \frac{1}{2} \right)^{n-1} \).

Step 4: Set the General Formula Equal to the Last Term

We know the last term is \( \frac{1}{1024} \). Set the general expression equal to this value: \[ 2 \left( \frac{1}{2} \right)^{n-1} = \frac{1}{1024} \].

Step 5: Solve for n

First, divide both sides by 2: \( \left( \frac{1}{2} \right)^{n-1} = \frac{1}{2048} \). Then recognize that \( \frac{1}{2048} = \left( \frac{1}{2} \right)^{11} \). Thus, \( n - 1 = 11 \). Solve for \( n \) to get \( n = 12 \).

Key Concepts

First Term

In a geometric sequence, the first term is the initial number from which the sequence begins. This term is critical because it forms the foundation for finding any term within the sequence. In our exercise, the first term is easily identified as the first number in the series provided:

• The sequence is given as: 2, 1, \( \frac{1}{2} \), ..., \( \frac{1}{1024} \).
• Thus, the first term, \( a_1 \), is \( 2 \).

Identifying the first term is the starting point for determining other essential elements such as the common ratio and developing the general formula.

Common Ratio

In a geometric sequence, the common ratio is the factor that each term is multiplied by to get the next term. It ensures the pattern is consistent across the sequence. Calculating the common ratio effectively uses any consecutive terms in the sequence:

• Take two successive terms; for instance, 1 and 2 in our sequence.
• To find the common ratio \( r \), divide the second term by the first: \( r = \frac{1}{2} \div 2 = \frac{1}{4} \).

The common ratio is essential for both continuing the sequence and deriving the general term formula.

General Term Formula

The general term formula of a geometric sequence provides a way to find any term in the sequence without listing them all. The formula expresses the \( n \)-th term as a function of the first term and the common ratio:
\( a_n = a_1 \cdot r^{n-1} \)
Inserting our known values, we derive:

• First term \( a_1 = 2 \).
• Common ratio \( r = \frac{1}{4} \).
• The formula becomes \( a_n = 2 \cdot \left( \frac{1}{4} \right)^{n-1} \).

This formula allows you to calculate any term in the sequence quickly. It is pivotal, especially when you need to find terms far down the sequence without computing each one individually.

Finite Sequence

A finite sequence is a list of numbers where the number of terms is limited. In our exercise, to find how many terms are in the finite geometric sequence, we use the general term formula and set it equal to the last term:
The equation becomes:
\[ 2 \left( \frac{1}{4} \right)^{n-1} = \frac{1}{1024} \] Solving this involves:

• The steps begin with dividing both sides by the first term 2 to get: \( \left( \frac{1}{4} \right)^{n-1} = \frac{1}{2048} \).
• Recognizing \( \frac{1}{2048} \) as \( \left( \frac{1}{4} \right)^{11} \), equate \( n-1 = 11 \).
• Solving for \( n \) gives \( n = 12 \).

This tells us the sequence has 12 terms, demonstrating the power of the general term formula in determining the length of a finite sequence.