Problem 61 - Algebra and Trigonometry

Question

If the numbers \(a_{1}, a_{2}, \ldots, a_{n}\) form a geometric sequence, then \(a_{2}, a_{3}, \ldots, a_{n-1}\) are geometric means between \(a_{1}\) and \(a_{n} .\) Insert three geometric means between 5 and \(80 .\)

Step-by-Step Solution

Answer

The three geometric means are 10, 20, and 40.

Step 1: Identify the General Form of a Geometric Sequence

A geometric sequence is defined as a sequence of numbers where each term after the first is the product of the previous term and a constant called the common ratio, \( r \). We begin with the first term \( a_1 = 5 \) and the last term \( a_5 = 80 \), which includes the three geometric means we need to find.

Step 2: Apply the Formula for the nth Term of a Geometric Sequence

The formula for the \( n \)-th term of a geometric sequence is given by \( a_n = a_1 \cdot r^{(n-1)} \). Here, \( a_5 = a_1 \cdot r^4 = 80 \).

Step 3: Calculate the Common Ratio

We solve the equation \( 80 = 5 \cdot r^4 \) to find \( r \). Dividing both sides by 5 gives \( r^4 = 16 \). Taking the fourth root of both sides, we find \( r = \sqrt[4]{16} = 2 \).

Step 4: Insert the Geometric Means

Using the common ratio \( r = 2 \), calculate the three geometric means:\- The first intermediate term: \( a_2 = a_1 \cdot r = 5 \times 2 = 10 \)\- The second intermediate term: \( a_3 = a_2 \cdot r = 10 \times 2 = 20 \)\- The third intermediate term: \( a_4 = a_3 \cdot r = 20 \times 2 = 40 \).

Step 5: Verify the Sequence

Ensure that the sequence \( 5, 10, 20, 40, 80 \) forms a geometric sequence. Check that each term is obtained by multiplying the previous term by the common ratio, \( r = 2 \). The sequence follows the pattern, confirming that the calculations are correct.

Key Concepts

Geometric Means

Geometric means serve as the intermediate values in a geometric sequence. When you insert geometric means between two numbers, they form a smoother transition, keeping the multiplication factor constant throughout. Imagine you have the numbers 5 and 80, and you want to fill in three numbers in between. These numbers are the geometric means.

To do this, you calculate each successive term by multiplying the previous term by a constant ratio. This ensures that all the numbers share a harmonious relationship, maintaining a consistent growth pattern.

• The process starts with the initial number (5 in this example).
• By continuously applying the common ratio, you find each geometric mean.
• This results in a sequence: 5, 10, 20, 40, 80, where 10, 20, and 40 are the geometric means.

Understanding how to calculate and place geometric means helps in multiple fields, from modeling growth trends to structuring finance calculations.

Common Ratio

The common ratio is a critical component in understanding geometric sequences. It's the constant number that you multiply each term by to progress to the next term. This ratio ensures the sequence's pattern and consistency.

For example, in the sequence from 5 to 80, we identified the common ratio as 2 by using the formula for geometric progression. The equation involved applying the nth term formula: \[a_5 = a_1 \cdot r^{4} = 80\]From here, by simplifying and solving for \( r \), you find:\[r^4 = 16\]and therefore, \[r = \sqrt[4]{16} = 2\].
This shared ratio of 2 allows you to start with 5 and calculate the precise values of 10, 20, 40, seamlessly leading up to 80, with each being double the previous term.

Frequently working with common ratios revolutionizes how sequences are constructed, directly impacting areas that are built upon repetitive scaling processes.

Nth Term Formula

The nth term formula is a powerful tool allowing us to find any term in a geometric sequence instantly. The formula is expressed as:\[a_n = a_1 \cdot r^{(n-1)}\]where \(a_n\) is the term you're trying to find, \(a_1\) is the first term in the sequence, \(r\) is the common ratio, and \(n\) is the term number.

It's particularly useful when you're not simply listing all terms, as it gives the ability to jump directly to the term you need without repetitive calculations.

• For instance, in our task of finding the sequence between 5 and 80, the formula helped us to directly calculate \(a_5\) by plugging in our known values.
• By knowing \(a_5 = 80\), \(a_1 = 5\), and the common ratio \(r\) found through earlier calculations, it efficiently leads to verify any term in the sequence.

This formula is essential for more complex sequences or when dealing with large \(n\), where manually stepping through each term is inefficient. Mastering the nth term formula puts you ahead in dealing with exponential growth or decay scenarios.