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Chapter 3: Problem 66

Let \(n \in N, n>25 .\) Let \(A, G, H\) denote the arithmetic mean, geometric mean and harmonic mean of 25 and \(n .\) The least value of \(n\) for which \(A, G, H \in\\{25,26, \ldots, n\\}\) is a. 49 b. 81 c. 169 d. 225

Short Answer

Expert verified
The least value of \( n \) is 169.

Step by step solution

01

Understanding Means

First, we should understand what arithmetic mean (A), geometric mean (G), and harmonic mean (H) are. For two numbers 25 and \( n \), they are defined as follows: - The arithmetic mean \( A \) is \( \frac{25 + n}{2} \).- The geometric mean \( G \) is \( \sqrt{25 \times n} \).- The harmonic mean \( H \) is \( \frac{2 \times 25 \times n}{25 + n} \).
02

Analyze Arithmetic Mean

To ensure that \( A \) is an integer between 25 and \( n \), solve: \[ \frac{25 + n}{2} = k \] for some integer \( k \) such that \( 25 \leq k \leq n \). Rearranging, we get: \[ n = 2k - 25 \].
03

Analyze Geometric Mean

For \( G \) to be an integer, \( \sqrt{25 \times n} \) must be a perfect square. Let \( 25 \times n = m^2 \) where \( m \) is an integer, implying that \( n = \frac{m^2}{25} \) and \( m \) should be a multiple of 5 for \( n \) to be integer.
04

Analyze Harmonic Mean

Ensure that \( H \) is an integer: \[ \frac{50n}{25+n} = h \].Multiply through by \( 25+n \): \[ 50n = h(25 + n) \].Solve for \( n \): \[ n = \frac{25h}{50-h} \]. \( n \) must be an integer satisfying this equation.
05

Find Least n

The least \( n \) that satisfies all of these conditions is when \( n = \frac{m^2}{25} \), \( m=25 \) (since \( G \) should also be integer).Hence, \( n = \frac{25^2}{25} = 25 \). But \( n > 25 \), so check next perfect square: \( m = 30 \). Thus, \( n = \frac{30^2}{25} = 36 \), still not \( > 49 \). Next, let \( m = 35 \), \( n = \frac{35^2}{25} = 49 \), which is the smallest \( n \).
06

Validate Conditions for n = 49

Check conditions: - Arithmetic Mean: \( A = \frac{25+49}{2} = 37 \), which is an integer.- Geometric Mean: \( G = \sqrt{25\times49} = \sqrt{1225} = 35 \), which is an integer.- Harmonic Mean: \( H = \frac{2 \times 25 \times 49}{25+49} = \frac{2450}{74} = 33.11 \approx 33 \), which is not an integer. Go to next step.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Mean
The arithmetic mean is an average calculated by adding a set of numbers and then dividing the sum by the count of those numbers. For instance, if the two numbers are 25 and \( n \), the arithmetic mean \( A \) can be calculated using the formula:
  • \( A = \frac{25 + n}{2} \)
This formula finds the middle value of 25 and \( n \) by evenly distributing the sum. To ensure \( A \) is an integer within the range \([25, n]\), determine an integer \( k \) so that \( 25 \leq k \leq n \). Rewriting for \( n \), the expression becomes:
  • \( n = 2k - 25 \)
Understanding this setup helps in efficiently evaluating if numbers fit within the defined criteria when calculating the arithmetic mean.
Geometric Mean
Geometric mean is another way to determine the average of a set of numbers, particularly beneficial when dealing with products or exponential growth. For numbers 25 and \( n \), it is calculated by taking the square root of their product:
  • \( G = \sqrt{25 \times n} \)
In order for \( G \) to be an integer, \( \sqrt{25 \times n} \) must itself be an integer, implying that \( 25 \times n \) should be a perfect square. Therefore, one can express \( 25 \times n \) as \( m^2 \), where \( m \) is an integer and a multiple of 5 (to ensure \( n \) stays an integer). Solving for \( n \), results in:
  • \( n = \frac{m^2}{25} \)
This approach ensures that the geometric mean fits operational constraints, mainly integer values and the range among the given data points.
Harmonic Mean
The harmonic mean offers a way to find averages that is useful when the numbers are rates or ratios. For example, regarding the numbers 25 and \( n \), the harmonic mean \( H \) is computed as:
  • \( H = \frac{2 \times 25 \times n}{25 + n} \)
To ensure that \( H \) is an integer, the calculation involves simplifying the expression so it divides evenly with an integer solution. The equation translates into:
  • \( 50n = h(25 + n) \)
Solving further, we can determine \( n \) as:
  • \( n = \frac{25h}{50 - h} \)
The harmonic mean hence must be logically evaluated to confirm it satisfies conditions set within the context of number operations, balancing the input parameters to achieve a mean that acts as an integer.

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Most popular questions from this chapter

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