Problem 123 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find all the numbers \(x\) and \(y\) and such that \(x, x+2 y, 2 x+y\) from an A.P. while the numbers \((y+1)^{2}\), \(x y+5,(x+1)^{2}\) form a G.P. Write down the progressions.

Step-by-Step Solution

Answer

The solutions are \(x = 4, y = 4\) and \(x = -1, y = -1\). The progressions are A.P.: 4, 12, 12 and G.P.: 25, 20, 25 for \(x = 4, y = 4\) and A.P.: -1, -3, -3 and G.P.: 4, -4, 4 for \(x = -1, y = -1\).

Step 1 - Find Relationship from A.P.

Three numbers \(x, x+2y, 2x+y\) form an A.P. if the common difference is the same. This means: \(x+2y - x = 2x+y - (x+2y)\). Solve this equation to get \(y = x\).

Step 2 - Find Relationship from G.P.

Three numbers \((y+1)^{2}, xy+5, (x+1)^{2}\) form a G.P. if the common ratio is the same. This means: \((xy+5) / (y+1)^{2} = (x+1)^{2} / (xy+5)\). Replace \(y\) with \(x\) (from Step 1) and simplify to get \(x^2 -3x - 4 = 0\).

Step 3 - Solve the Quadratic Equation

Solve the equation \(x^2 -3x - 4 = 0\) which is a simple quadratic equation with roots \(x = 4\) or \(x = -1\). Substitute these into \(y = x\) to get corresponding \(y\) values.

Step 4 - Write Down the Progressions

Using the values of \(x\) and \(y\), write down the A.P. and G.P. sequences.

Key Concepts

Quadratic Equations

Quadratic equations play a critical role in mathematics, especially when tackling problems involving algebraic expressions. A quadratic equation is a second-degree polynomial equation in a single variable, typically presented in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The solutions to these equations are called roots, which can be found using various methods:

• The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Factoring the quadratic expression
• Completing the square

In our provided solution, the quadratic to solve was \(x^2 - 3x - 4 = 0\). By factoring, the roots of this equation were found as \(x = 4\) or \(x = -1\). These roots provide solutions for the original problem in terms of the values for \(x\) and \(y\). Understanding the properties of quadratic equations can greatly simplify problem-solving in algebra.

Sequences and Series

Sequences and series form the backbone of understanding progression patterns in mathematics. A sequence is an ordered list of numbers, while a series is the sum of a sequence of terms. Two special kinds of sequences are Arithmetic Progressions (A.P.) and Geometric Progressions (G.P.).

• Arithmetic Progressions: In an A.P., the difference between consecutive terms, known as the common difference, is constant. For example, if \(x, x+2y, 2x+y\) form an A.P., we discovered that their differences are equal, leading to the condition \(y = x\).
• Geometric Progressions: In a G.P., the ratio between consecutive terms, known as the common ratio, is constant. In our problem, the numbers \((y+1)^2, xy+5, (x+1)^2\) were found to be in a G.P., which further provided an equation that required solving.

Understanding these progressions and their properties is essential in defining relationships and solving series-related problems seamlessly.

Problem Solving in Algebra

Algebra provides tools for expressing relationships through equations and inequalities, making problem-solving structured and systematic. The key problem-solving strategy is identifying patterns or relationships within the problem. Breaking it into manageable steps ensures a smooth solution process.
Let's recap the approach in our example:

• Step 1: Start by identifying the pattern or property, such as A.P. or G.P., and set up conditions accordingly.
• Step 2: Translate these conditions into equations. Use given information to derive relationships between the variables involved.
• Step 3: Solve these equations. This often includes solving linear or quadratic equations, as demonstrated with \(x^2 - 3x - 4 = 0\).
• Step 4: Substitute back to find the desired variables or progressions.

This organized approach simplifies complex problems by tackling them in digestible parts, ensuring clarity and accuracy in problem-solving activities.