Problem 75
Question
If \(0.272727 \ldots, x\) and \(0.727272 \ldots\) are in H.P., then \(x\) must be (A) rational (B) integer (C) irrational (D) None of these
Step-by-Step Solution
Verified Answer
The correct answer is (A) rational.
1Step 1: Understand Harmonic Progression (H.P.)
In a Harmonic Progression, the reciprocals of the terms form an Arithmetic Progression (A.P.). For three terms \( a, b, c \) to be in H.P., their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) must be in A.P.
2Step 2: Identify Recurring Decimals
Given that the first term is \(0.272727\ldots\), which is the recurring decimal \(\frac{3}{11}\), and the third is \(0.727272\ldots\), which is \(\frac{8}{11}\). These terms are represented as fractions when dealing with sequences.
3Step 3: Use the Relation of H.P.
Since \( a = \frac{3}{11}, b = x, \) and \( c = \frac{8}{11} \) are in H.P., their reciprocals must form an A.P.: \( \frac{1}{\frac{3}{11}}, \frac{1}{x}, \frac{1}{\frac{8}{11}} \). This simplifies to \( \frac{11}{3}, \frac{1}{x}, \frac{11}{8} \) in A.P.
4Step 4: Set up the Arithmetic Progression Equation
For \( \frac{11}{3}, \frac{1}{x}, \frac{11}{8} \) to be in A.P., the difference between any two consecutive terms must be the same. Therefore, \( \frac{1}{x} - \frac{11}{3} = \frac{11}{8} - \frac{1}{x} \).
5Step 5: Solve the Equation
Simplify and solve the equation: \( \frac{1}{x} - \frac{11}{3} = \frac{11}{8} - \frac{1}{x} \). Add \( \frac{1}{x} \) to both sides to get \(2 \times \frac{1}{x} = \frac{11}{3} + \frac{11}{8}\). Continue by finding a common denominator for \(\frac{11}{3}\) and \(\frac{11}{8}\), which is 24, leading to \( \frac{88}{24} + \frac{33}{24} = \frac{121}{24} \). Thus, \( \frac{2}{x} = \frac{121}{24} \).
6Step 6: Calculate \(x\)
Solving for \(x\) gives the equation \( x = \frac{2 \times 24}{121} = \frac{48}{121} \). This is a rational number.
Key Concepts
Recurring DecimalsArithmetic ProgressionSolving Equations
Recurring Decimals
Recurring decimals are numbers with infinite repeating digits. For example, numbers like 0.272727... are called recurring decimals since the digits 27 repeat indefinitely. Such numbers can be converted into fractions. If a decimal has one repeating part, like our example, it can be turned into a fraction by setting the decimal equal to a variable, multiplying the variable to shift the repeating part, and then subtracting and solving for the variable. In this case, with 0.272727..., we let it equal to a variable, solve it to convert into the fraction \( \frac{3}{11} \).
Reciprocal conversion plays a significant role when these numbers are used in harmonic progressions, making it essential to recognize and convert recurring decimals accurately.
Reciprocal conversion plays a significant role when these numbers are used in harmonic progressions, making it essential to recognize and convert recurring decimals accurately.
Arithmetic Progression
Arithmetic Progression (A.P.) is a sequence of numbers where the difference between any two consecutive terms is constant. This constant is called the common difference.
For instance, if you have terms like 2, 5, 8, 11, you notice the common difference is 3.
When dealing with harmonic progressions, the reciprocals of the terms form an arithmetic progression, as seen in our case. For terms \(\frac{11}{3}\), \(\frac{1}{x}\), and \(\frac{11}{8}\) to form an A.P., the difference between any two consecutive reciprocal terms must be constant.
For instance, if you have terms like 2, 5, 8, 11, you notice the common difference is 3.
When dealing with harmonic progressions, the reciprocals of the terms form an arithmetic progression, as seen in our case. For terms \(\frac{11}{3}\), \(\frac{1}{x}\), and \(\frac{11}{8}\) to form an A.P., the difference between any two consecutive reciprocal terms must be constant.
Solving Equations
Solving equations involves finding the value of an unknown that makes the equation true. In our problem, solving for \(x\) required manipulating an equation derived from the A.P. condition.
- First, set up the equation based on the arithmetic progression condition: \( \frac{1}{x} - \frac{11}{3} = \frac{11}{8} - \frac{1}{x} \).
- Simplify by combining like terms to isolate \( \frac{1}{x} \): \( 2 \times \frac{1}{x} = \frac{11}{3} + \frac{11}{8} \).
- To solve, find the common denominator to add fractions on the right side, obtaining \( \frac{121}{24} \).
- Resolve for \(x\) to get its value: \( x = \frac{48}{121} \). This solution shows \(x\), a rational number, fits the conditions set by the problem.
Other exercises in this chapter
Problem 72
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