Problem 17 - Algebra 2 and Trigonometry

Question

Find four arithmetic means between 3 and \(18 .\)

Step-by-Step Solution

Answer

The four arithmetic means are 6, 9, 12, and 15.

Step 1: Understand the Problem

The problem asks us to find four arithmetic means between the numbers 3 and 18. This means we need a sequence of numbers where each pair of consecutive numbers has a constant difference, called the common difference. Thus, the sequence will have a total of 6 terms: the first term (3), four arithmetic means, and the last term (18).

Step 2: Determine the Arithmetic Sequence Components

An arithmetic sequence is defined by its first term and the common difference. Here, the first term is 3, and the last term is 18. Let the common difference be denoted by \(d\). The sequence will be: \(3, \, (3+d), \, (3+2d), \, (3+3d), \, (3+4d), \, 18\).

Step 3: Formulate the Equation

Since 18 is the last term in the sequence, we can write this in terms of the initial term and the number of terms. Using the arithmetic sequence formula for the nth term, \(a_n = a_1 + (n-1)\cdot d\), the 6th term is 18. Hence, \(3 + 5d = 18\).

Step 4: Solve for the Common Difference

Rearrange the equation from Step 3 to solve for \(d\): \(5d = 18 - 3\), which simplifies to \(5d = 15\). Dividing both sides by 5, we find \(d = 3\).

Step 5: Find the Arithmetic Means

With \(d = 3\), substitute \(d\) back into the sequence to find the terms. The sequence becomes: \(3, \, 3 + 3, \, 3 + 2\cdot3, \, 3 + 3\cdot3, \, 3 + 4\cdot3, \, 18\), which simplifies to \(3, \, 6, \, 9, \, 12, \, 15, \, 18\). The arithmetic means are the second, third, fourth, and fifth terms of this sequence: 6, 9, 12, and 15.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a fundamental concept that helps in forming the sequence of numbers. It is the constant difference between any two consecutive terms in the sequence. This uniformity allows you to easily predict and calculate future terms once you know the first term and the common difference.

To determine the common difference, you can subtract any term from the one that follows it. For example, if you have a sequence like 2, 5, 8, the common difference \(d\) is calculated as follows: 5 - 2 = 3 and 8 - 5 = 3. The consistent result confirms that the common difference \(d\) is 3.

Identifying the common difference is crucial when you need to fill in missing terms or extend the sequence. It becomes especially handy in problems where you are given the first and last terms alongside a requirement to find additional intermediate terms, as seen in the exercise provided.

Arithmetic Mean

An arithmetic mean in the context of an arithmetic sequence refers to the intermediate values located between two known terms in the sequence. When tasked with "finding arithmetic means," the goal is to identify the numbers that complete a continuous set where each pair of neighboring terms has the same common difference.

Consider the solution to the exercise of finding four arithmetic means between 3 and 18. In this scenario, the arithmetic means are the four numbers that complete the sequence: 6, 9, 12, and 15. Each of these numbers fits seamlessly into the sequence due to the consistent common difference of 3.

Understanding arithmetic means is important for grasping how sequences can be used to model real-world situations and trends, providing a foundation for more advanced mathematical concepts such as percentages and growth rates.

Sequence Formula

The sequence formula is an essential tool for working with arithmetic sequences. It allows one to find any term in a sequence if the first term and the common difference are known. The general sequence formula is expressed as:\[a_n = a_1 + (n-1) \cdot d\]Here, \(a_n\) represents the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

By substituting the appropriate values into this formula, you can solve for unknowns or confirm your sequence. In the exercise, the formula helps to determine the common difference and eventually the arithmetic means. For instance, with the sequence beginning at 3 and ending at 18, using the formula for the sixth term \(6\):\[a_6 = 3 + 5d\]Setting \(a_6\) to 18 gives:\[3 + 5d = 18\]Solving for \(d\) gives the common difference, which then leads to finding all the intermediate values in the sequence effectively.

Utilizing this formula, one can confidently tackle varying problems involving arithmetic sequences, as it provides a structured method for calculating and verifying sequence terms with precision.