Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{3}=50, a_{7}=0.005$$

Step-by-Step Solution

Answer

The first term \(a_1\) is 5000 and the common ratio \(r\) is 0.1.

Step 1: Understand the Problem

We are given that the third term of a geometric sequence, denoted as \(a_3\), is 50, and the seventh term, \(a_7\), is 0.005. We need to find the first term \(a_1\) and the common ratio \(r\) of this geometric sequence.

Step 2: Use the Formula for Geometric Sequences

The general term of a geometric sequence can be found using the formula \(a_n = a_1 \, r^{n-1}\). For \(a_3\), we have \(a_3 = a_1 \, r^2 = 50\). For \(a_7\), we have \(a_7 = a_1 \, r^6 = 0.005\).

Step 3: Set Up Equations

Using the formula from Step 2, we set up two equations: 1. \(a_1 \cdot r^2 = 50\)2. \(a_1 \cdot r^6 = 0.005\)

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

Divide the second equation by the first equation to eliminate \(a_1\):\(\frac{a_1 \cdot r^6}{a_1 \cdot r^2} = \frac{0.005}{50}\) which simplifies to \(r^4 = 0.0001\).Taking the fourth root of both sides yields \(r = 0.1\).

Step 5: Find \(a_1\) Using \(r\)

Substitute \(r = 0.1\) back into the first equation, \(a_1 \cdot (0.1)^2 = 50\). This gives \(a_1 \cdot 0.01 = 50\), so \(a_1 = \frac{50}{0.01} = 5000\).

Step 6: Conclude the Solution

The first term \(a_1\) is 5000 and the common ratio \(r\) is 0.1. These values satisfy the conditions of the geometric sequence given in the problem.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key element that helps determine the behavior of the sequence. It is the factor by which each term is multiplied to get the next term. Once you know the common ratio, you can determine any term if you have the first term. To find the common ratio, especially when the terms are non-consecutive, it’s important to use the formula involving given terms.

In our problem, we know:

• \( a_3 = a_1 \cdot r^2 = 50 \)
• \( a_7 = a_1 \cdot r^6 = 0.005 \)

By dividing one equation by the other and simplifying, we were able to isolate and solve for \( r \).

• Division gives \( r^4 = 0.0001 \)
• Solving for \( r \), the fourth root of 0.0001 is 0.1

This ratio of 0.1 is what we multiply each term by to get the next in the sequence.

Geometric Progression Formula

The geometric progression formula is the essential tool for finding terms within a geometric sequence. The formula is:\[ a_n = a_1 \cdot r^{n-1}\] Where:

• \( a_n \) is the \( n \)-th term,
• \( a_1 \) is the first term of the sequence,
• \( r \) is the common ratio,
• and \( n \) is the term number.

This formula allows us to express each term based on its position \( n \) and the well-defined elements \( a_1 \) and \( r \).

Using the formula in our exercise, we were able to set up the sequence's terms with two equations, helping us unlock the values of the first term \( a_1 \) and the common ratio \( r \). It effectively connects any term back to the start of the sequence, reinforcing how each element is inherently linked.

Term Calculation

Calculating specific terms in a geometric sequence involves using the information you already have - particularly the first term and the common ratio. This enables us to follow the progression and calculate subsequent terms easily.

In our exercise, after finding that the common ratio \( r \) is 0.1 and using the formula for that sequence, we calculated:\[ a_1 = \frac{a_3}{r^2} = \frac{50}{(0.1)^2} = 5000.\]This backward calculation involved using known terms to deduce \( a_1 \), which is essential for understanding the entire sequence. Once we have \( a_1 \), calculating any further term is straightforward.

• You can find any term by continuing to apply \( a_n = a_1 \cdot r^{n-1} \).
• Each calculation depends upon prior terms and the consistent ratio through the sequence.

This step-by-step calculation methodical approach ensures that each term in the sequence is accurately determined based on foundational elements.