Problem 236 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(x, 1, z\) are in AP and \(x, 2, z\) are in GP, then find the harmonic mean of \(x\) and \(z\).

Step-by-Step Solution

Answer

Under the assumption that x and z are real numbers, there are no valid solutions for the given exercise and therefore the harmonic mean of x and z cannot be found.

Step 1: Use AP relationship

Since \(x, 1, z\) are in AP, we can write: \(x + z = 2 \cdot 1\) This simplifies to: \( x + z = 2 \)

Step 2: Use GP relationship

Since \(x, 2, z\) are in GP, we can write: \(2^2 = x \cdot z\) This simplifies to: \( 4 = xz \)

Step 3: Solve equations for x and z

We now have a system of linear equations: \( x + z = 2 \) \( xz = 4 \) Solve the first equation for either x or z. In this case, we will solve for x: \( x = 2 - z \) Substitute this expression for x into the second equation: \( (2 - z)z = 4 \) Expand and simplify: \( 2z - z^2 = 4 \) Rearrange the equation into a quadratic equation: \( z^2 - 2z + 4 = 0 \)

Step 4: Solve the quadratic equation

To find the values of z that satisfy this equation, we can use the quadratic formula: \[ z = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(4)}}{2(1)} \] Simplify and calculate: \[ z = \frac{2 \pm \sqrt{-12}}{2} \] Since we have a negative value inside the square root, we only have complex solutions for z. However, it's assumed that x and z should be real numbers. This means that there is no valid solution for this given exercise. In conclusion, under the assumption that x and z should be real numbers, there is no valid solution to finding the harmonic mean of x and z for this exercise.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. In a simple formula, if the first term is denoted by 'a' and the common difference by 'd', then the nth term \(a_n\) can be expressed as \(a_n = a + (n - 1)d\).

To identify terms that are in an AP, the relationship between consecutive terms can be checked. If the sequence meets the condition that the difference between successive terms is constant, the sequence is considered arithmetic. In the context of the exercise provided, when we have three numbers where the second is the arithmetic mean of the first and third, the relationship \(x + z = 2 \times 1\) establishes that \(x, 1, z\) are in an AP with common difference zero.

Geometric Progression

A geometric progression (GP), also known as a geometric sequence, is defined by a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. If 'a' is the first term and 'r' is the common ratio, then the nth term \(a_n\) is given by \(a_n = a \cdot r^{(n-1)}\).

In practice, to determine if numbers form a GP, divide successive terms and see if the result is constant. For instance, in the problem at hand, given that \(x, 2, z\) are in a GP, we derive the relationship \(2^2 = x \cdot z\) to express the fact that 2 is the geometric mean of \(x\) and \(z\), hence confirming they follow a GP with a common ratio of \(2/x\).

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, which means they include a term with the variable raised to the power of two. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. These equations can be solved for the variable using methods like factoring, completing the square, or using the quadratic formula: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\].

As illustrated in the solution steps, we encounter a quadratic equation when we combine the relationships derived from the AP and GP conditions. Solving this equation is crucial for finding the values of \(x\) and \(z\) that satisfy both conditions. However, the presence of a negative number under the square root, also known as the discriminant, implies the solutions will be complex and not real numbers.

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part, and they're represented in the form \(a + bi\), where 'a' is the real part, 'b' is the imaginary part, and 'i' is the square root of -1. They extend the idea of the one-dimensional number line into the two-dimensional complex plane, allowing for the solution of equations that have no real solution, such as those with negative discriminants in the quadratic formula.

In our context, when solving the quadratic equation from the AP and GP relationships, we encountered a negative discriminant, resulting in complex solutions. This reveals an important property of the quadratic equation: when the discriminant \(b^2 - 4ac\) is less than zero, the solutions to \(ax^2 + bx + c = 0\) enter the realm of complex numbers. Since the original problem seeks real number solutions for \(x\) and \(z\), the complex solutions indicate an impossibility to find a real harmonic mean in this scenario.