Problem 53 - A Complete Resource Book in Mathematics for JEE Main

Question

If three positive numbers \(a, b, c\) are in H.P., then \(a^{n}+c^{n}\) \((\mathrm{A})>2 b^{n}\) \((\mathrm{B})=2 b^{n}\) \((\mathrm{C})<2 b^{n}\) \((\mathrm{D})>b^{n}\)

Step-by-Step Solution

Answer

(A) \(a^n + c^n > 2b^n\)

Step 1: Defining Harmonic Progression

If three numbers \(a, b, c\) are in Harmonic Progression (H.P), it implies that their reciprocals \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) are in Arithmetic Progression (A.P). Therefore, we can write the A.P condition as \(2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c}\).

Step 2: Simplify the harmonic condition

Using the A.P condition, we have:\[ \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \]\[ \frac{2}{b} = \frac{a+c}{ac} \]Cross-multiplying gives:\[ 2ac = b(a+c) \]Express \(a+c\) in terms of \(b\):\[ a+c = \frac{2ac}{b} \]

Step 3: Analyze the expression \(a^n + c^n\)

Notice, \(a\) and \(c\) are symmetric around \(b\) due to the Harmonic Mean property. In other words, since \(a, b, c\) are in H.P., typically \(b\) is less than both \(a\) and \(c\), so \(a > b\) and \(c > b\).

Step 4: Assess the inequality \( a^n + c^n \) versus \(2b^n\)

Given that \(a\) and \(c\) are greater than \(b\) and symmetric (since \(b\) is the harmonic mean), their powers \(a^n\) and \(c^n\) will also be greater than \(b^n\). Thus, \(a^n + c^n > 2b^n\).

Step 5: Conclusion

Thus, the relation for the values of \(a^n + c^n\) compared to \(2b^n\) given that exactly fits the condition is \(a^n + c^n \gt 2b^n\).

Key Concepts

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is often called the "common difference." For example, in the sequence 2, 4, 6, 8, each term increases by 2, making 2 the common difference.
In the context of Harmonic Progression, if we have three numbers that are in A.P., such as the reciprocals 1/a, 1/b, and 1/c, this implies:

• The sequence has a common difference, i.e., \( \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b} \).
• This relationship can then be expressed as: \( 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \).

This setup is crucial to understanding Harmonic Progressions, as it allows us to relate the terms with a simple arithmetic approach. By solving this equation, we gain insights into the behavior of terms in both Arithmetic and Harmonic Progressions.

Harmonic Mean

The Harmonic Mean is a type of average, useful in contexts where rates and ratios are important. It is defined as the reciprocal of the arithmetic mean of the reciprocals of a given set of numbers.
The formula for the Harmonic Mean of two numbers, a and b, is:

• \( HM = \frac{2}{\frac{1}{a} + \frac{1}{b}} \)
• For three numbers like in a Harmonic Progression, like a, b, and c, the Harmonic Mean of a and c is conveniently the middle term, b.

This characteristic is vital in our exercise as it highlights the symmetric property of the numbers in the progression. When dealing with Harmonic Progressions, the main takeaway is how the "middle" number (b in our exercise) acts as an average note of the outer numbers.

Inequality in Progressions

Inequalities frequently arise in sequences and progressions, showing how numbers relate under particular conditions. In this problem, we analyze the inequality between powers: \( a^n + c^n \) and \( 2b^n \).
Key to our problem:

• Our numbers, a and c, are larger than b when they form a Harmonic Progression, suggesting \( a^n > b^n \) and \( c^n > b^n \).
• Combining these, it follows that \( a^n + c^n \) naturally exceeds \( 2b^n \).

This inequality is intuitive as it results from the symmetric placement of the numbers around b, reinforcing their "outward" growth compared to the mean. Recognizing such patterns aids in solving complex mathematical relationships.

Symmetric Property of Means

Symmetric means indicate a balance around a central value. This property becomes apparent in progressions, such as A.P.s and H.P.s, where terms can align symmetrically around an average or central term.
For Harmonic Progressions, the symmetry explains:

• The "balance" where b, as the Harmonic Mean, equally divides the set into two parts.
• This means a and c are reflections, and deviations from b are mirror images: \( a - b = c - b \) when considering their reciprocal equivalents.

This symmetry is exactly why the initial conditions in the exercise hold — a pattern consistent across mean types. Embracing the symmetric nature in these sequences aids in swift masterminding of solutions, especially when faced with complex inequality challenges.