Problem 56 - College Algebra with Modeling and Visualization

Question

Find a general term \(a_{n}\) for the geometric sequence. $$a_{2}=6, a_{4}=24, r>0$$

Step-by-Step Solution

Answer

The general term is \( a_n = 3 \times 2^{n-1} \).

Step 1: Understand the Geometric Progression Formula

The general term of a geometric sequence is given by the formula \(a_n = a_1 imes r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Step 2: Create Equations Using Given Terms

From the problem, we know \(a_2 = 6\) and \(a_4 = 24\). Using the general formula, we can write:1. \(a_2 = a_1 imes r = 6\)2. \(a_4 = a_1 imes r^3 = 24\).

Step 3: Solve for the Common Ratio \(r\)

Divide the second equation by the first one: \[ \frac{a_4}{a_2} = \frac{a_1 imes r^3}{a_1 imes r} = r^2 \]\[ \frac{24}{6} = r^2 \]\[ r^2 = 4 \]Since \(r > 0\), \(r = 2\).

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

Use the equation \(a_2 = a_1 imes r = 6\):\[ a_1 imes 2 = 6 \]\[ a_1 = \frac{6}{2} = 3 \]

Step 5: Find the General Term \(a_n\)

Substitute \(a_1\) and \(r\) back into the general term formula:\[ a_n = a_1 imes r^{(n-1)} = 3 imes 2^{(n-1)} \].

Key Concepts

General Term Formula

A geometric sequence, often termed as a geometric progression, is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Understanding how to find the general term formula of a geometric sequence is critical in identifying any term's value from the sequence without having to find all the preceding terms.

The general term formula for a geometric sequence is given by:

• \( a_n = a_1 \times r^{(n-1)} \)

Here:

• \( a_n \) represents the nth term of the sequence,
• \( a_1 \) is the first term,
• \( r \) is the common ratio, and
• \( n \) denotes the term number in the sequence.

This formula precisely calculates the nth term by continually applying the multiplication of the first term and the powers of the common ratio. In the exercise, substituting the found values gives:\[ a_n = 3 \times 2^{(n-1)} \] This relationship aids in predicting any term of the sequence instantly, without iterative calculations.

Common Ratio

The common ratio \( r \) in a geometric sequence is the multiplier that links successive terms. Recognizing this ratio is crucial as it holds the sequence together, allowing each term to smoothly scale from the one before it.

In instructional terms, to find \( r \), you can divide any term in the sequence by its preceding term:\[ r = \frac{a_{n}}{a_{n-1}} \]Using our original numbers, to find the common ratio, we implement:

• For given terms, \( r^2 = \frac{24}{6} = 4 \).
• Since \( r > 0 \), solve \( r = \sqrt{4} \).
• The common ratio is found to be \( r = 2 \).

This easy computation reveals how the sequence's terms grow or shrink, and in our sequence, each term is simply double the previous one.

First Term Calculation

Finding the first term \( a_1 \) of a geometric sequence is often essential when other terms are given. The whole sequence can be rebuilt once the first term and the common ratio are confirmed.

From the exercise, given \( a_2 = 6 \) and knowing \( r = 2 \), we can find \( a_1 \) fairly straightforwardly:

• Use the relationship \( a_2 = a_1 \times r \).
• Substitute known values: \( 6 = a_1 \times 2 \).
• Solve for \( a_1 \): \( a_1 = \frac{6}{2} = 3 \).

This simple algebraic manipulation provides the starting point of the sequence. Knowing both \( a_1 \) and \( r \) assists in determining the sequence fully, which can then be utilized in various real-life applications or additional mathematical analyses.