Problem 78 - Precalculus: Mathematics for Calculus

Question

The arithmetic mean (or average) of two numbers \(a\) and \(b\) is $$m=\frac{a+b}{2}$$ Note that \(m\) is the same distance from \(a\) as from \(b,\) so \(a, m, b\) is an arithmetic sequence. In general, if \(m_{1}, m_{2}, \ldots, m_{k}\) are equally spaced between \(a\) and \(b\) so that $$a, m_{1}, m_{2}, \ldots, m_{k}, b$$ is an arithmetic sequence, then \(m_{1}, m_{2}, \ldots, m_{t}\) are called \(k\) arithmetic means between \(a\) and \(b\) (a) Insert two arithmetic means between 10 and 18 . (b) Insert three arithmetic means between 10 and 18 . (c) Suppose a doctor needs to increase a patient's dosage of a certain medicine from 100 mg to 300 mg per day in five equal steps. How many arithmetic means must be inserted between 100 and 300 to give the progression of daily doses, and what are these means?

Step-by-Step Solution

Answer

(a) 12.67, 15.34; (b) 12, 14, 16; (c) 140, 180, 220, 260.

Step 1: Understanding the Arithmetic Sequence

To find arithmetic means between two numbers, the sequence of numbers between them must be equally spaced. In an arithmetic sequence, the difference between consecutive terms is constant, called the common difference.

Step 2: Determining the Common Difference for (a)

For (a), to insert two arithmetic means between 10 and 18, the sequence has 4 terms in total (10, m1, m2, 18). The sequence's common difference, \(d\), can be calculated as follows:\[d = \frac{b-a}{k+1} = \frac{18-10}{2+1} = \frac{8}{3}\approx 2.67 .\]

Step 3: Finding the Arithmetic Means for (a)

Starting at 10, add the common difference \(d\) to find each subsequent term:1. \(m_1 = 10 + d = 10 + 2.67 \approx 12.67\)2. \(m_2 = m_1 + d \approx 12.67 + 2.67 \approx 15.34\)So the two arithmetic means are approximately 12.67 and 15.34.

Step 4: Determining the Common Difference for (b)

For (b), to insert three arithmetic means between 10 and 18, the sequence has 5 terms in total (10, m1, m2, m3, 18). The common difference \(d\) is:\[d = \frac{18-10}{3+1} = \frac{8}{4} = 2.\]

Step 5: Finding the Arithmetic Means for (b)

For the sequence 10, m1, m2, m3, 18 with \(d = 2\), determine each mean by:1. \(m_1 = 10 + 2 = 12\)2. \(m_2 = 12 + 2 = 14\)3. \(m_3 = 14 + 2 = 16\)The three arithmetic means are 12, 14, and 16.

Step 6: Understanding Problem (c)

The doctor needs to increase the dosage from 100 mg to 300 mg in five equal steps. This means finding four arithmetic means between 100 and 300.

Step 7: Determining the Common Difference for (c)

For problem (c), the sequence includes 6 terms (100, m1, m2, m3, m4, 300). Therefore, the common difference \(d\) is:\[d = \frac{300-100}{4+1} = \frac{200}{5} = 40.\]

Step 8: Finding the Arithmetic Means for (c)

For the sequence 100, m1, m2, m3, m4, 300 with \(d = 40\), calculate each mean:1. \(m_1 = 100 + 40 = 140\)2. \(m_2 = 140 + 40 = 180\)3. \(m_3 = 180 + 40 = 220\)4. \(m_4 = 220 + 40 = 260\)The four arithmetic means are 140, 180, 220, and 260.

Key Concepts

Arithmetic Means

Arithmetic means refer to numbers that are inserted between two given numbers in such a way that the resulting sequence is an arithmetic sequence. This means that each number, including the inserted ones, is equally spaced from each other.
To better understand, consider two numbers, say 10 and 18. If we want to insert arithmetic means between them, we want to ensure that these means are evenly distanced, forming a balanced progression.

• For example, if inserting two arithmetic means, the terms would appear as: 10, m1, m2, 18.
• To ensure each number is equally spaced, we calculate the common difference and use it to determine the positions of m1 and m2.

In the context of the exercise, the arithmetic means provide a simple yet powerful illustration of how numbers can be added to sequences to maintain equal spacing between them.

Common Difference

An essential element of any arithmetic sequence is the common difference, denoted by \(d\). This difference is what you add to each term to get to the next one in the sequence. It’s the "gap" that remains constant between any two consecutive numbers in the sequence.
To find the common difference between two numbers where several arithmetic means need to be inserted in between, the formula is:\[ d = \frac{b-a}{k+1} \]where \(a\) and \(b\) are the given numbers, and \(k\) is the number of arithmetic means to be added.

• This formula ensures that when you insert the arithmetic means between two numbers, each number in the sequence is equally spaced by this common difference.
• In our examples, for finding three arithmetic means between 10 and 18, the common difference is determined to be 2.

Knowing the common difference allows students to construct sequences methodically, one step at a time, ensuring the integrity of the arithmetic sequence.

Linear Progression

A linear progression is another term often used interchangeably with an arithmetic sequence or series. It refers to a sequence of numbers where the difference between any two successive terms is the same. This equal spacing creates a "linear" look when graphed, resembling a straight line.
The beauty of a linear progression lies in its predictability and simplicity. It doesn’t matter which two consecutive terms you pick: the difference remains the same, thus maintaining a consistent pattern.

• The process of determining a linear progression involves calculating the common difference and using it to seamlessly bridge the gap between initial and final terms with arithmetic means.
• This concept is practically applicable, such as in dosing schedules where medicine needs to be increased gradually in a consistent manner.

The importance of linear progression in real-life situations is significant—it aids in creating systematic plans and predictions based on consistent increments.