Question

If \(\mathrm{A}, \mathrm{G}, \mathrm{H}\) are the AM, GM and HM between any two distinct positive real numbers, then out of the following 4 Statements (i) \(\mathrm{G}^{2}=\mathrm{AH}\) (ii) \(\mathrm{A}>\mathrm{G}>\mathrm{H}\) (iii) \(\mathrm{A}=\mathrm{G}=\mathrm{H}\) (iv) \(\mathrm{G}=\mathrm{A}^{2} \mathrm{H}\) (a) Only (i) and (iii) are correct (b) Only (ii) is correct (c) Only (i) and (ii) are correct (d) All are correct

Short answer

a) Only (i) is correct b) Only (i) and (iii) are correct c) Only (i) and (ii) are correct d) Only (ii) and (iii) are correct Answer: c) Only (i) and (ii) are correct.

Step-by-step solution

Understanding the AM, GM, and HM of two positive real numbers

The formulas for the means are: Arithmetic Mean (AM), A = (a + b) / 2 Geometric Mean (GM), G = √(ab) Harmonic Mean (HM), H = 2ab / (a + b) where a and b are distinct positive real numbers.

Prove the relationship between the means

First, we can write G² = ab, which simplifies one of the given statements. Now, we'll start with proving the relationship between AM, GM, and HM. To prove the inequality, we'll use the AM-GM inequality: AM = (a + b) / 2 ≥ √(ab) = GM GM ≥ HM = 2ab / (a + b) The inequality between GM and HM can be proven by: (ab) / (a + b) ≤ √(ab) This inequality can also be written as: a²b² ≤ (a + b)²(ab) After rearranging the terms, we get: a²b² ≤ a²b + ab² Dividing all sides by a²b², we get: 1 ≤ (a / b) + (b / a) Here, GM² = ab, which basically implies statement (i) is correct. So, statement (i) is true. Since G² = AH (which indicates that the algebraic relationship between G, A, and H is correct) statement (iii) cannot be true since A ≠ G ≠ H.

Checking the validity of the remaining statements

Now, we need to determine if statement (ii) and (iv) are true. Statement (ii): From the AM-GM-HM relationship in Step 2, we know that A > G > H. So, statement (ii) is true. Statement (iv): By comparing G² = ab with statement (iv), G = A²H, we see that this statement doesn't make sense and contradicts the relationship found in Step 2. In conclusion, the analysis shows that only statements (i) and (ii) are true, and thus the correct answer is: (c) Only (i) and (ii) are correct.

Key concepts

Arithmetic Mean (AM)

The Arithmetic Mean, commonly referred to as the average, plays an integral role in statistics and real-world applications. When we have two distinct positive real numbers, let's call them a and b, their Arithmetic Mean (AM) is calculated by simply adding them together and dividing by two. The formula looks like this:
\[ AM = \frac{a + b}{2} \]
In essence, the AM represents a midpoint in the range of the numbers, providing a measure of central tendency. When learning about AM, it's helpful to picture it as the balance point on a scale where a and b are weights at each end. This visualization aids in understanding why AM is a figure of spread, representing how two numbers are spaced out on a number line.

Geometric Mean (GM)

Differing from the Arithmetic Mean, the Geometric Mean is a method to ascertain the central tendency of a set of numbers by considering their product rather than their sum. For two distinct positive real numbers a and b, the Geometric Mean is the square root of their product. It's given by the formula:
\[ GM = \sqrt{ab} \]
The GM is particularly useful when dealing with numbers that represent rates of change, such as growth rates in biology or finance. Since the GM accounts for the compounding effect, it offers a more accurate measure in these scenarios than the AM. A useful tip when approaching problems involving GM: recognize that geometric means tend to be less affected by extreme values in a data set, unlike arithmetic means.

Harmonic Mean (HM)

Harmonic Mean stands out when we're looking at rates and their respective times. This mean is most valuable when average rates are to be calculated, such as speed. For two distinct positive real numbers a and b, their Harmonic Mean (HM) is given by:
\[ HM = \frac{2ab}{a + b} \]
To understand why the HM is used for rates, consider two segments of a journey at different speeds. The HM provides the overall average speed for the entire journey when time, rather than distance, is constant for each segment. It's crucial to note that the HM is always the lowest among the AM, GM, and HM, reflecting the reciprocal nature of rates in its formula.

AM-GM-HM Inequality

One of mathematics' fundamental inequalities is the AM-GM-HM Inequality. This principle asserts that for any set of positive real numbers, their Arithmetic Mean is always greater than or equal to their Geometric Mean, which in turn is always greater than or equal to their Harmonic Mean. Symbolically, for any two positive real numbers a and b, we can write it as:
\[ AM \geq GM \geq HM \]
The equality holds true only when all the numbers in the set are the same, meaning a = b. The inequality helps in various mathematical proofs and is used frequently in optimization problems. Remember, this inequality reflects the balance of relationships among means and can be leveraged to solve complex mathematical issues, particularly in competitive exams like the IIT-JEE.