Problem 64 - Algebra and Trigonometry

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 $$a, m_{1}, m_{2}, \ldots, m_{k}, b$$ 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 \(\mathrm{mg}\) to 300 \(\mathrm{mg}\) 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) Means: 12.67, 15.33\n(b) Means: 12, 14, 16\n(c) Means: 140 mg, 180 mg, 220 mg, 260 mg

Step 1: Understanding the Problem

To find arithmetic means between two numbers, we need to know that they are parts of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant, known as the common difference.

Step 2: Finding Common Difference for Two Means (a)

Given: Sequence is 10, _ , _, 18. There are two arithmetic means.Number of intervals (gaps) = 3,Let the common difference be \(d\).\(d = \frac{18 - 10}{3} = \frac{8}{3} = 2.67\) approximately.

Step 3: Calculate Two Means (a)

Start with 10 and add \(d\) to find each subsequent mean.10 + \(d\) \(\approx 12.67\), 12.67 + \(d\) \(\approx 15.33\), Therefore, the two arithmetic means are 12.67 and 15.33.

Step 4: Finding Common Difference for Three Means (b)

Given: Sequence is 10, _ , _, _, 18. There are three arithmetic means.Number of intervals (gaps) = 4,Let the common difference be \(d\).\(d = \frac{18 - 10}{4} = 2\).

Step 5: Calculate Three Means (b)

Start with 10 and add \(d = 2\) to find each subsequent mean.10 + 2 = 12,12 + 2 = 14,14 + 2 = 16,Therefore, the three arithmetic means are 12, 14, and 16.

Step 6: Determining Means for Dosage Increase (c)

Given: Sequence is 100, ..., ..., ..., ..., ..., 300. The total number of steps required is 5 equal steps in total. This means there are 4 arithmetic means.Let the common difference be \(d\).\(d = \frac{300 - 100}{5} = 40\).

Step 7: Calculate Dosage Steps (c)

Start with 100 and add \(d = 40\) to find each subsequent mean.100 + 40 = 140,140 + 40 = 180,180 + 40 = 220,220 + 40 = 260,Therefore, the four steps (arithmetic means) are 140 mg, 180 mg, 220 mg, and 260 mg per day.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. Imagine you are climbing stairs, where each step is evenly spaced apart. Numbers in an arithmetic sequence follow this idea closely. By using a consistent step size or 'common difference,' each number moves a set distance from the one before and grows sequentially.

• First Term (usually denoted as \( a \)) – This is where the sequence starts.
• Common Difference (\( d \)) – This is the "step size" across the sequence.
• General Formula – Any term in the sequence can be found by: \( a_n = a + (n-1)d \).

Sequences are crucial for systematic progression and they’re simple to understand when you think of them as literally "stepping" from one number to the next, with each step being the same length.

Common Difference

The common difference in an arithmetic sequence is the amount added (or subtracted if negative) to each term to get to the next one. It's what guarantees that the sequence remains evenly spaced.For example, if given the sequence 3, 7, 11, 15,..., the common difference \(d\) here is 4:

• Used to calculate new terms through addition.
• Derived from: \( d = a_{n} - a_{n-1} \), where \( a_n \) is any term and \( a_{n-1} \) is its predecessor.
• In our problem example, for inserting means between numbers, \(d\) determines the exact steps in sequence accordingly.

Think of \(d\) as the key that unlocks the door to evenly distributed numbers, enabling precision in predicting sequence terms.

Intervals in Arithmetic Series

Intervals in an arithmetic series refer to the "spaces" or "gaps" between numbers when inserted between two limits. Counting the intervals provides insight into the possible number of new terms added. Consider the sequence formed: 10, _, _ ,18. Here:

• The number of intervals = The number of spaces available + 1. For two means: three gaps (two means).
• Allows calculation of arithmetic means through \( d = \frac{b-a}{n+1} \), where \( n \) is the number of inserted means.
• Forms the bridge connecting each subsequent number clearly and evenly.

The concept of intervals ensures that adding new numbers means maintaining uniform spacing.

Dosage Progression Using Arithmetic Sequence

When managing medical dosages, arithmetic sequences can effectively design safe and controlled progressions. For example, if a doctor aims to adjust medicine from 100 mg to 300 mg:

• It can be split into equal steps based on arithmetic sequences.
• Identify total steps and required increments to manage adjustments smoothly, like the solution given for increasing dose: 100, ..., ..., ..., 300 requires four means with \( d = 40 mg \).
• These intervals ensure that the dosage increases at a steady and manageable rate.

Application of arithmetic series here ensures both precision and safety, enabling the doctor to easily monitor and control dosage increments.