Question

\(\mathrm{x}_{1} \mathrm{x}_{2} ; \mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{z}_{1}, \mathrm{z}_{2}\) are the two AM's, two GM's, two HM's respectively between a and b. \(\mathrm{G}_{1}\) is the GM between \(\mathrm{y}_{1}\) and \(y_{2^{*}} G_{2}\) is the GM between \(z_{1}\) and \(z_{2} ; A_{1}\) is the AM between \(x_{1}\) and \(x_{2}\) and \(A_{2}\) is the AM between \(z_{1}\) and \(z_{2}\). Then \(A_{1}: A_{2}=\) (a) \(\mathrm{G}_{1}: \mathrm{G}_{2}\) (b) \(\mathrm{G}_{2}: \mathrm{G}_{1}\) (c) \(\mathrm{G}_{1}^{2}: \mathrm{G}_{2}^{2}\) (d) \(\mathrm{G}_{2}{ }^{2}: \mathrm{G}_{1}{ }^{2}\)

Short answer

Question: Find the ratio of arithmetic means A1 and A2, where A1 is the arithmetic mean between the two arithmetic means x1 and x2 between a and b, and A2 is the arithmetic mean between the two harmonic means z1 and z2 between a and b. The given options are (a) G1:G2, (b) G2:G1, (c) G1^2:G2^2, and (d) G2^2:G1^2, where G1 is the geometric mean between the two geometric means y1 and y2 between a and b, and G2 is the geometric mean between the two harmonic means z1 and z2 between a and b. Answer: (d) G2^2:G1^2

Step-by-step solution

Express the AM's and GM's in terms of a and b

Since \(x_1, x_2\) are two AM's between a and b, we have: \(x_1 = a + \frac{b-a}{3}\), \(x_2 = a + 2\frac{b-a}{3}\) Since \(y_1, y_2\) are two GM's between a and b, we have: \(y_1 = a^{\frac13}b^{\frac23}\), \(y_2 = a^{\frac23}b^{\frac13}\) %

Express the HM's in terms of a and b

Since \(z_1, z_2\) are two HM's between a and b, we have: \(z_1 = \frac{ab}{a+2(b-a)}\), \(z_2 = \frac{ab}{2a+ (b-a)}\) Now let's find the arithmetic means \(A_1\) and \(A_2\):

Find A1 and A2

Arithmetic mean between \(x_1\) and \(x_2\): \(A_1 = \frac{x_1 + x_2}{2} = \frac{a + \frac{b-a}{3} + a + 2\frac{b-a}{3}}{2} = \frac{2a + b}{3}\) Arithmetic mean between \(z_1\) and \(z_2\): \(A_2 = \frac{z_1 + z_2}{2} = \frac{\frac{ab}{a+2(b-a)} + \frac{ab}{2a+ (b-a)}}{2} = \frac{ab}{2(a+b)}\)

Calculate G1 and G2

Geometric mean between \(y_1\) and \(y_2\): \(G_1 = \sqrt{y_1 \cdot y_2} = \sqrt{a^{\frac13}b^{\frac23} \cdot a^{\frac23}b^{\frac13}} = \sqrt{ab}\) Geometric mean between \(z_1\) and \(z_2\): \(G_2 = \sqrt{z_1 \cdot z_2} = \sqrt{\frac{ab}{a+2(b-a)} \cdot \frac{ab}{2a+ (b-a)}} = \sqrt{\frac{a^2b^2}{(a+b)^2}}= \frac{ab}{a+b}\)

Find the ratio A1 : A2

Now we have all the elements needed to find the ratio \(A_1 : A_2\): \(A_1 : A_2 = \frac{2a + b}{3} : \frac{ab}{2(a+b)}\) Comparing \(A_1\) and \(A_2\) with \(G_1\) and \(G_2\), we can observe that: \(A_1 = \frac{2G_2^2}{3}\) and \(A_2 = \frac{G_2^2}{2}\) Thus, on simplification \(A_1 : A_2 = \frac{2G_2^2}{3} : \frac{G_2^2}{2}\) Therefore, \(A_1 : A_2 = G_2^2 : G_1^2\) So, the correct option is (d) \(G_2^2 : G_1^2\).

Key concepts

Mean Value Theorems

Mean Value Theorems are essential concepts in calculus and mathematics that provide a link between the values of a function over an interval. They help determine averages and give insights into the function's behavior across that interval. In this exercise, we use different types of mean values, primarily focusing on Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM). These means are used to bridge the properties of two numbers, here denoted as "a" and "b".

• Arithmetic Mean (AM): The arithmetic mean is the simple average of two numbers. For this exercise, if we have two arithmetic means between "a" and "b", they are calculated by evenly dividing the interval between them.
• Geometric Mean (GM): This mean represents the central tendency of a set of numbers by using the product of their values. Geometric means in this context are leveraged to find the midpoint of multiplication between the numbers "a" and "b".
• Harmonic Mean (HM): The harmonic mean is used for rates or ratios, calculated as the reciprocal of the arithmetic mean of the reciprocals of the given numbers.

Understanding these means helps grasp the relationships between sets of numbers and their ratios better. The exercise essentially involves computing these means and deriving solutions from their properties and interactions.

Sequence and Series

Sequences and series are fundamental concepts in mathematics that deal with ordered lists of numbers. In this exercise, the focus is on arithmetic, geometric, and harmonic sequences and how we can generate means from these sequences.

• Sequential Relationships: A sequence is a set of numbers arranged in a specific order, each member is called a term. Understanding the logic of progression allows the calculation of AM, GM, and HM from specific sequences, exemplified by the numbers between given terms here.
• Progression in Series: Series is essentially the sum of the sequence. In arithmetic and geometric contexts, calculating the means from series assists in understanding the overall behavior of the series across its domain.

In the exercise, we calculate AM, GM, and HM and conclude with relationships between these classes of means. This helps in building advanced logical skills required for deeper mathematical problem-solving.

Mathematical Ratio Comparisons

In mathematics, comparing ratios helps us understand the relative sizes of two quantities. This exercise involves comparing the ratios of two computed means, associated with AM and GM, to derive a specific result.

• Ratio Calculation: A ratio expresses how many times one number contains another. It's a powerful comparative tool in mathematics. Here, we calculated the ratio of arithmetic means compared to geometric means as AM: GM to find conclusions about their equivalent expressions.
• Use of Ratios in Solutions: By comparing the ratios of different means — such as G2 and G1 — to determine that certain relationships hold true (e.g., G2: G1 = G2^2: G1^2), it allows for a comparison that simplifies complex mathematical expressions.

Understanding the role of ratios in mathematics is key. Here, using the ratio helped conclude the exercise by matching the determined relationship of two major types of means. This problem illustrates the powerful use of ratios in deducing meaningful results in mathematical expressions.