Problem 20 - Algebra for JEE ADVANCED

Question

The next term of the G.P. \(x, x^{2}+2\), and \(x^{3}+10\) is a. \(\frac{729}{16}\) b. 6 c. 0 d. 54

Step-by-Step Solution

Answer

54

Step 1: Identify the Common Ratio of G.P.

In a geometric progression (G.P.), the common ratio between consecutive terms is constant. If the first term is \(a_1 = x\) and the second term is \(a_2 = x^2 + 2\), then the common ratio \(r\) is given by \(r = \frac{a_2}{a_1} = \frac{x^2 + 2}{x}\). Similarly, the common ratio between the second term and the third term is \(r = \frac{a_3}{a_2} = \frac{x^3 + 10}{x^2 + 2}\).

Step 2: Equate Common Ratios

Equate the expressions for the common ratio obtained in Step 1: \(\frac{x^2 + 2}{x} = \frac{x^3 + 10}{x^2 + 2}\). Cross-multiply to obtain \((x^2 + 2)^2 = x(x^3 + 10)\).

Step 3: Solve the Equation

Expand \((x^2 + 2)^2\) to get \(x^4 + 4x^2 + 4\). Also, distribute \(x\) in \(x(x^3 + 10)\) to get \(x^4 + 10x\). Thus, the equation becomes \(x^4 + 4x^2 + 4 = x^4 + 10x\). Simplify by canceling \(x^4\) from both sides: \(4x^2 + 4 = 10x\). This simplifies to \(4x^2 - 10x + 4 = 0\).

Step 4: Solve the Quadratic Equation

The quadratic equation \(4x^2 - 10x + 4 = 0\) can be solved using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = -10\), and \(c = 4\). Calculate the discriminant: \(b^2 - 4ac = (-10)^2 - 4(4)(4) = 100 - 64 = 36\). Thus, the solutions are \(x = \frac{10 \pm \sqrt{36}}{8}\) which gives \(x = \frac{16}{8}\) (or 2) and \(x = \frac{4}{8}\) (or \(\frac{1}{2}\)).

Step 5: Determine the Next Term of the G.P.

Substitute \(x = 2\) into the G.P. to find the next term. The current terms are \(2, 6, 18\). Calculate the common ratio: \(r = \frac{6}{2} = 3\). To find the next term, multiply the last term by \(r = 3\), thus \(18 \times 3 = 54\).

Key Concepts

Common Ratio in Geometric Progression

In a geometric progression (G.P.), the common ratio is the factor by which we multiply one term to get to the next term. This ratio is constant throughout the sequence. For example, in a sequence like \(a, ar, ar^2, ar^3, \ldots\), the common ratio \(r\) can be found by dividing any term by its preceding term.
Let's consider a practical example from our exercise: we have terms \(x, x^2 + 2\), and \(x^3 + 10\). You'll find the common ratio by dividing the second term by the first (\(\frac{x^2+2}{x}\)) and ensuring it matches the ratio between the third and second terms (\(\frac{x^3 + 10}{x^2 + 2}\)).
This consistency in the ratio is crucial for the series to qualify as a geometric progression and allows us to predict future terms based on the established pattern.Understanding the common ratio is key in solving problems involving geometric sequences, like predicting unknown terms or verifying a sequence's geometric nature.

Solving a Quadratic Equation

Often, problems involving sequences might lead us to solving quadratic equations, as seen in solving the common ratio condition for our geometric series. A quadratic equation is of the form \(ax^2 + bx + c = 0\).
This kind of equation can be solved using factors, completing the square, or more commonly, the quadratic formula:

• Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our exercise, after simplifying the common ratio condition, we ended up with the quadratic equation \(4x^2 - 10x + 4 = 0\). Calculating the discriminant \(b^2 - 4ac\) helps in determining the nature of the roots.
For our example, the discriminant is 36, indicating real and distinct roots, leading to solutions \(x = 2\) and \(x = \frac{1}{2}\). With these solutions, we can confirm which value fits the requirements of our sequence problem best and use it to further compute terms in our series.

Sequences and Series

Sequences are ordered lists of numbers following a specific pattern, while a series is the sum of the terms in a sequence. In mathematics, distinguishing between the two is essential.
Geometric progressions, like the one in our exercise, are specific types of sequences where each term is obtained by multiplying the previous one by a constant factor, known as the common ratio. If you add the terms of a geometric sequence, it becomes a geometric series.
For instance, with the sequence \(2, 6, 18, 54, \ldots\), each term is three times the previous one, making it geometric. If such a sequence follows a recognizable pattern, predicting future terms or calculating the total of the series becomes manageable.
Understanding the defining characteristics of sequences and series, including how to manipulate and solve equations derived from them, is crucial for tackling problems in mathematics effectively.