Question

if \(\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}\), then \(a, b, c\) are in (a) AP (b) \(\mathrm{GP}\) (c) HP (d) AGP

Short answer

Based on the given equation, the variables a, b, and c form a Harmonic Progression (HP).

Step-by-step solution

Simplify the Given Equation

The given equation is: \( \frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c} \) To simplify it, first find a common denominator on both sides of the equation : \( \frac{(a-c)+(b-a)}{(b-a)(b-c)}= \frac{a+c}{ac} \) Keep simplifying: \( \frac{b+c}{(b-a)(b-c)}= \frac{a+c}{ac} \)

Compare Both Sides of the Equation

Now, we will compare both sides of the equation, cross-multiply, and simplify: \((b+c)(ac)= (a+c)(b-a)(b-c)\) Expanding the equation gives: \(abc+ac^2= (ab-a^2c-b^2c+ac^2)\) Subtracting \(ac^2\) and rearrange the terms: \(abc-ab= ac^2-b^2c+a^2c-ab\)

Factor the Resulting Equation

Now, factor the expression: \(ab(c-1) = ac(a+b)-b^2c(c-1)\)

Analyze the Resulting Equation

The resulting equation suggests a relationship between the terms \(a, b, c\): \( \frac{c-1}{a+b} = \frac{a(c-1)}{ab}= \frac{b(c-1)}{ab} \) Since the numerators are equal, we can say: \(\frac{1}{a+b} = \frac{1}{ab}\) Which means: \(a+b=ab\) Now we will see if this condition (\(a+b=ab\)) satisfies any of the four progressions:

Check for Arithmetic Progression (AP)

In an AP, there is a common difference (d) such that: \(b=a+d\) \(c=b+d\) Using this relation, we will try to check if our condition is satisfied: \(a+(a+d) = (a)(a+d)\) This does not provide a consistent relationship between a, b, and c. Therefore, a, b, c do not form an arithmetic progression.

Check for Geometric Progression (GP)

In a GP, there is a common ratio (r) such that: \(b=ar\) \(c=ar^2\) Using these relations, we will try to check if our condition is satisfied: \(a+(ar) = (a)(ar)\) \(a(1+r)= a^2r\) This does not provide a consistent relationship between a, b, and c. Therefore, a, b, c do not form a geometric progression.

Check for Harmonic Progression (HP)

In an HP, if we take the reciprocals of the terms, they form an AP: \(\frac{1}{a}\), \(\frac{1}{b}\), \(\frac{1}{c}\) are in AP. Meaning: \(b=\frac{2ac}{a+c}\) Using this relation, we will try to check if our condition is satisfied: \(a+\frac{2ac}{a+c} = a\left(\frac{2ac}{a+c}\right)\) \( (a+c)^2 = 2a^2c\) \(a^2c-2ac^2+c^3=0\) \(a+b=ab\) is satisfied. Therefore, a, b, and c form a harmonic progression.

Check for Arithmetic-Geometric Progression (AGP)

In an AGP, the terms form a sequence of the product of terms in an AP and a GP: \(a, b, c\) are both AP and GP. As we have previously checked, the terms do not form both AP and GP. Therefore, they do not form an arithmetic-geometric progression. So, based on this analysis, the answer is: The given variables a, b, c are in: **(c) Harmonic Progression (HP)**

Key concepts

Algebraic Equations

Algebraic equations play a crucial role in various areas of mathematics and are foundational tools for solving problems and establishing relationships between different quantities. An algebraic equation is a mathematical statement asserting the equality of two expressions, typically involving variables and constants. Let's take a closer look at how these work in practice:

• An algebraic equation usually involves operations such as addition, subtraction, multiplication, and division.
• These equations are used to describe various types of mathematical relationships or constraints.
• In solving algebraic equations, we often aim to find the values of unknown variables that satisfy the equation.

In the original exercise, we have an equation that involves three variables, namely, \(a\), \(b\), and \(c\). This equation is manipulated using algebraic techniques, such as cross-multiplication and simplification, to reveal underlying relationships between the variables. This follows the standard method of solving equations by isolating variables and reducing terms.

Sequence and Series

Sequences and series are foundational concepts in mathematics that describe ordered lists of numbers. A sequence is a set of numbers listed in a specific order, while a series is the sum of the terms of a sequence.

• Sequence: This is a function whose domain is the set of natural numbers, and each number is referred to as a term of the sequence.
• Series: If you take a sequence and add its terms together, you get a series. Each number in this sum is called a partial sum.
• Convergence: A series is said to converge if the sequence of partial sums tends to a limit.

In the context of the exercise, the sequence involves numbers \(a\), \(b\), and \(c\), and examining these as potential members of different types of sequences helps to identify any progression they might follow. This is done by applying standard tests for arithmetic, geometric, and harmonic progressions.

Mathematical Progression

Mathematical progressions are specific types of sequences where each term is derived from a specific rule or formula. Understanding these types can help solve complex equations and find patterns in sequences:

• Arithmetic Progression (AP): In AP, each term after the first is obtained by adding a constant difference to the previous term.
• Geometric Progression (GP): In GP, each term is obtained by multiplying the previous term by a constant ratio.
• Harmonic Progression (HP): In HP, the reciprocals of the terms form an Arithmetic Progression.

In the exercise, the goal was to determine if variables \(a\), \(b\), and \(c\) follow any of these progressions. By considering the relations and transforming the terms of the equation provided, it was proven that \(a\), \(b\), and \(c\) form a Harmonic Progression (where the reciprocals form an Arithmetic Progression). Recognizing these structures can greatly simplify the process of solving complex mathematical problems.