Problem 20 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\(a_1 = 5000\) and \(r = 0.1\).

Step 1: Formula for nth term of geometric sequence

For a geometric sequence, the nth term is given by the formula: \( a_n = a_1 imes r^{(n-1)} \). Here, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 2: Write equations for given terms

We have two terms from the sequence: \(a_3 = 50\) and \(a_7 = 0.005\). Using the formula, we can write:1. \( a_3 = a_1 imes r^{2} = 50 \).2. \( a_7 = a_1 imes r^{6} = 0.005 \).

Step 3: Express the second equation in terms of the first

Divide the second equation by the first equation to eliminate \(a_1\):\[ \frac{a_7}{a_3} = \frac{a_1 imes r^6}{a_1 imes r^2} \].This simplifies to:\( r^4 = \frac{0.005}{50} \).

Step 4: Solve for \(r\)

Calculate the ratio on the right-hand side:\( r^ \) corresponds to:\[ r^4 = \frac{0.005}{50} = 0.0001 \].Now, solve for \(r\) using the fourth root,\[ r = \sqrt[4]{0.0001} \].Thus, \( r = 0.1 \).

Step 5: Solve for \(a_1\)

Substitute the value of \(r\) back into the first equation:\( 50 = a_1 imes (0.1)^2 \).This simplifies to: \( 50 = a_1 imes 0.01 \).Now solve for \(a_1\):\( a_1 = \frac{50}{0.01} = 5000 \).

Step 6: Conclusion

The values of \(a_1\) and \(r\) are determined. \(a_1\) is 5000, and \(r\) is 0.1.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio, denoted as \( r \), is a consistent factor by which each term is multiplied to obtain the next term. This concept is crucial as it dictates the nature and behavior of the sequence.

To find the common ratio, you usually divide any term in the sequence by the previous term. However, if specific terms are given, like in our problem with \( a_3 = 50 \) and \( a_7 = 0.005 \), we can derive \( r \) by setting up and solving equations based on the nth term formula.

Here are a few key points about the common ratio:

• If \( r > 1 \), the sequence grows.
• If \( 0 < r < 1 \), the sequence shrinks.
• If \( r < 0 \), terms will alternate in sign, creating a fluctuating sequence.

Understanding the common ratio helps us anticipate the sequence's progression, a vital component in both arithmetic tasks and mathematical problem solving.

Nth Term Formula

The nth term formula of a geometric sequence is crucial for identifying specific terms without listing the entire sequence.

It is expressed as:
\[ a_n = a_1 \times r^{n-1} \]
In this formula:

• \( a_n \) is the nth term of the sequence.
• \( a_1 \) is the first term.
• \( r \) is the common ratio.
• \( n \) is the term number.

The formula allows the direct calculation of any term using the first term and the common ratio.

For example, in the original problem, to find specific terms:
- Use known values to write equations for \( a_3 \) and \( a_7 \).
- Solve these equations to get the values of \( a_1 \) and \( r \). This practical application is a typical exercise when learning sequences.

Sequences and Series

Sequences are ordered sets of numbers following a specific pattern. In a geometric sequence, each term is determined by multiplying the previous term by a fixed non-zero number, known as the common ratio.

This type of sequence is only one part of sequences and series, which also includes arithmetic sequences and other forms. While a sequence is simply a list, a series involves summing the terms of a sequence.

Key distinctions include:

• Arithmetic sequences add a constant value (difference) to get each term.
• Geometric sequences multiply by a constant factor (ratio).
• Series sum up terms, which can also be arithmetic or geometric.

Understanding the nature of sequences and series is vital for solving complex mathematical problems and can be applied to real-world scenarios ranging from finance to computer science.

Mathematical Problem Solving

Mathematical problem solving involves using logical thinking and precise calculations to find solutions to various problems.

In our geometric sequence problem, this involves several steps, such as:

• Understanding and identifying known variables and the equations that correlate with them.
• Applying algebraic manipulation to simplify and solve equations for unknowns like \( a_1 \) and \( r \).
• Verifying solutions by substituting back into the original equations.

Effective problem solving demands clear understanding and strategic application of mathematical principles.

Here’s a brief approach to solving similar problems:
- Define the variables and known values.
- Write down the mathematical relationship (like the nth term formula for sequences).
- Use logical steps to manipulate equations and resolve unknowns.
- Always double-check your calculations for accuracy.