Problem 37 - Contemporary Precalculus

Question

In Exercises \(31-38,\) use the given information about the geometric sequence \(\left\\{a_{n}\right\\}\) to find as and a formula for \(a_{n}\). $$\left.a_{1}=5, a_{7}=20 \quad \text { (assume that } r>0\right)$$

Step-by-Step Solution

Answer

Answer: The formula for the n-th term of the given geometric sequence is a_n = 5 * (\sqrt[6]{4})^(n-1).

Step 1: Identify the geometric sequence and given terms

We are given the first term of the geometric sequence, a₁ = 5, and the seventh term a₇ = 20. We know that the common ratio (r) is greater than 0. Recall that the formula for the n-th term of a geometric sequence is: a_n = a₁ * r^(n-1)

Step 2: Find the common ratio r using the given terms

Using the formula for the n-th term of a geometric sequence, we can write the equation for the seventh term: a₇ = a₁ * r^(7-1) We know that a₇ = 20 and a₁ = 5, so we can plug in these values: 20 = 5 * r^6 Now, we will solve for r: r^6 = 20/5 r^6 = 4 Taking the 6th root of both sides, we get: r = \sqrt[6]{4} Since r > 0, we only consider the positive root.

Step 3: Find the formula for a_n

Now that we have found the common ratio r = \sqrt[6]{4}, we can write the formula for the n-th term of the geometric sequence: a_n = a₁ * r^(n-1) Substitute the values for a₁ and r: a_n = 5 * (\sqrt[6]{4})^(n-1) This is the formula for the n-th term of the given geometric sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a crucial component that determines how each term relates to the one before it. The common ratio, often represented by \( r \), is found by dividing any term of the sequence by its preceding term. In our exercise, we were given the first term, \( a_1 = 5 \), and the seventh term, \( a_7 = 20 \). To find \( r \), we used the formula for the \( n \)-th term:

• \( a_n = a_1 \cdot r^{(n-1)} \)

We plugged in the known values to form the equation \( 20 = 5 \cdot r^6 \). Solving this equation involves dividing both sides by 5 to get \( r^6 = 4 \), and then taking the sixth root of both sides. The result is \( r = \sqrt[6]{4} \). Since \( r > 0 \), we only consider the positive value of this root, which serves as our common ratio for the sequence.

n-th Term Formula

The \( n \)-th term formula in a geometric sequence allows us to find any term in the sequence without having to write out all the previous terms. This formula is particularly useful for sequences with many terms. The general formula is:

• \( a_n = a_1 \cdot r^{(n-1)} \)

In our specific problem, we've determined the first term \( a_1 = 5 \) and the common ratio \( r = \sqrt[6]{4} \). By substituting these values into the formula, we arrive at:

• \( a_n = 5 \cdot (\sqrt[6]{4})^{(n-1)} \)

This formula provides a systematic way to determine the value of any term \( a_n \) in this sequence, enabling us to predict future terms and analyze the sequence's behavior.

Sequences and Series

Sequences and series are mathematical concepts used to understand patterns and sums in numbers. A sequence, in particular, is an ordered list of numbers, following a specific rule. In the case of geometric sequences, each term is derived by multiplying the previous term by a fixed number known as the common ratio.
Geometric sequences are defined by their exponential nature, which makes them powerful models for various real-world phenomena, like population growth and investment compounding.
A series, on the other hand, involves summing the terms of a sequence. While our current focus is on sequences, understanding both concepts is vital for higher-level math, as they provide foundational knowledge for calculus and higher algebra.

• Geometric Sequence Example: \( 5, 5\sqrt[6]{4}, 5(\sqrt[6]{4})^2, \ldots \)
• Series Example: \( 5 + 5\sqrt[6]{4} + 5(\sqrt[6]{4})^2 + \ldots \)

Understanding the principles of sequences and series can open the door to a deeper appreciation of mathematical patterns and their applications.