Problem 64 - Precalculus: Mathematical for Calculus

Question

Arithmetic Means 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 $$a, m_{1}, m_{2}, \dots, m_{k}, b$$ is an arithmetic sequence, then \(m_{1}, m_{2}, \ldots, m_{k}\) 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, 15.33; (b) 12, 14, 16; (c) 140, 180, 220, 260.

Step 1: Understanding the Problem

We are asked to find arithmetic means between two numbers so that they form an arithmetic sequence. We need to determine these means for different scenarios provided.

Step 1: Calculate Two Arithmetic Means

Given that there are two arithmetic means (a, m1, m2, b) inserted between 10 and 18, this creates an arithmetic sequence with 4 terms. Use the common difference formula for an arithmetic sequence where d is the common difference. The formula is: \[ m_i = a + id \] for i = 1 to k. Calculate d using the formula \[ d = \frac{b - a}{k+1} \] where k is the number of means. So, \[ d = \frac{18 - 10}{3} = \frac{8}{3}\]. Thus, m1 = 10 + \frac{8}{3} and m2 = 10 + 2\times\frac{8}{3}, which results in m1 = \frac{38}{3} \approx 12.67 and m2 = 18 - \frac{8}{3} = \frac{46}{3} \approx 15.33.

Step 2: Calculate Three Arithmetic Means

Now, insert three arithmetic means (a, m1, m2, m3, b) in the sequence. Here, k = 3. Therefore, \[ d = \frac{18 - 10}{4} = 2. \] Calculate each mean: for m1, m2, and m3, m1 = 10 + 2\times1, m2 = 10 + 2\times2, m3 = 10 + 2\times3. Resulting in m1 = 12, m2 = 14, and m3 = 16.

Step 3: Dosage Calculation in Five Steps

The patient’s dosage must increase from 100 mg to 300 mg in five equal steps; hence we have to find four arithmetic means (a, m1, m2, m3, m4, b). Here, k = 4, so \[ d = \frac{300 - 100}{5} = 40. \] The four means are: m1 = 100 + 40\times1 = 140, m2 = 100 + 40\times2 = 180, m3 = 100 + 40\times3 = 220, and m4 = 100 + 40\times4 = 260.

Key Concepts

Arithmetic Mean

The arithmetic mean is a fundamental concept in mathematics, often referred to as the 'average.' When dealing with two numbers, the arithmetic mean is calculated by summing the numbers and dividing the total by two.

This concept extends well to arithmetic sequences. In such sequences, the arithmetic mean between two numbers not only signifies the average; it also maintains an equal distance from each of the original numbers, effectively placing it in the middle.

• Formula: The arithmetic mean between two numbers, say \(a\) and \(b\), is given by \(m = \frac{a + b}{2}\).
• Example: For numbers 10 and 18, the mean is \(m = \frac{10 + 18}{2} = 14\).

In problems requiring multiple arithmetic means between numbers, the concept is extended to mean that these means are equally spaced.
Thus, they form an arithmetic sequence where each term after the first is increased by a constant, known as the common difference.

Common Difference

The 'common difference' is a pivotal aspect of arithmetic sequences. It refers to the consistent interval between consecutive terms of the sequence, reflecting a uniform addition or reduction from one term to the next. Understanding and calculating this difference is essential when inserting arithmetic means.

The formula to determine the common difference ( \(d\)) between terms in an arithmetic sequence is:

• \(d = \frac{b - a}{k + 1}\), where \(a\) is the first term, \(b\) is the last term, and \(k\) is the number of arithmetic means inserted between \(a\) and \(b\).

For example, when inserting two arithmetic means between 10 and 18:

• Calculate \(d = \frac{18 - 10}{3} = \frac{8}{3}\).
• This calculates the equal step taken from one term to the next in the sequence.

Common difference is crucial for identifying all terms in a sequence.

Sequence Progression

Understanding sequence progression is key to deciphering how arithmetic sequences unfold. This concept elucidates the pattern and order by which each term follows the previous, structured by the common difference.

In an arithmetic sequence, the progression is defined as adding the common difference successively to each term. It's this linear progression that helps in constructing a sequence:

• If the first term is \(a\), the second term would be \(a + d\), the third \(a + 2d\), so forth.

Consider a case inserting three arithmetic means between 10 and 18. The sequence would follow:

• First term \(a = 10\)
• Second term \(a + d = 12\)
• Third term \(a + 2d = 14\)
• Fourth term \(a + 3d = 16\)
• Last term \(b = 18\)

Each term's placement is intrinsically tied to the common difference and clearly demonstrates the progression from start to finish.