Problem 26 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha, \beta\) are the roots of the equation \(375 x^{2}-25 x-2=0\) and \(S_{n}=\alpha^{n}+\beta^{n}\), then \(\operatorname{Lt}_{n \rightarrow \infty} \sum_{r=1}^{n} S_{r}\) is (A) \(\frac{7}{12}\) (B) \(\frac{1}{12}\) (C) \(\frac{35}{12}\) (D) None of these

Step-by-Step Solution

Answer

The limit is \(\frac{1}{12}\). Option (B).

Step 1: Recognize the Quadratic Equation

Given the quadratic equation \(375x^2 - 25x - 2 = 0\), identify that \(\alpha\) and \(\beta\) are its roots. To find the behavior of \(S_n = \alpha^n + \beta^n\), it's helpful to examine the properties of the roots.

Step 2: Use Vieta's Formulas

By Vieta's formulas, for a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots \(\alpha + \beta\) is \(-\frac{b}{a}\) and the product \(\alpha\beta = \frac{c}{a}\). Here \(\alpha + \beta = \frac{1}{15}\) and \(\alpha\beta = -\frac{2}{375}\).

Step 3: Apply the Recurrence Relation for \(S_n\)

The sequence \(S_n\) satisfies the recurrence relation derived from the quadratic equation: \(S_{n} = \frac{1}{15}S_{n-1} - \frac{2}{375}S_{n-2}\). This is because \(\alpha^n\) and \(\beta^n\) individually satisfy the same second-order linear recurrence relation as does the sequence \(S_n\).

Step 4: Analyze the Long-Term Behavior of \(S_n\)

Since the roots \(\alpha\) and \(\beta\) are from a quadratic with real coefficients but smaller absolute values in terms of their geometric progression, as \(n\) approaches infinity, both \(\alpha^n\) and \(\beta^n\) diminish towards zero faster.

Step 5: Evaluate the Sum \(\sum_{r=1}^{n} S_{r}\) and Its Limit

Given the rapid decay of \(S_n\), approximate the series using the finite terms that remain significant, noting that the sum's contribution sharply reduces with larger \(n\). The leading behavior derives from initial terms.

Step 6: Use Patterns and Approximation for Limit Evaluation

Observing and calculating the sum for explicit small values reveals the dominant initial few \(S_n\). Recognize that the contributions progressively sum towards the consistent pattern evaluation, limiting toward a fraction \(S_r\) converges to.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in mathematics, characterized by their standard form: \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). Quadratic equations can describe parabolic graphs and have a maximum of two roots, which can be real or complex. The roots are the values of \(x\) that make the equation true when substituted back in. Understanding these solutions is essential, as they offer insights into optimization problems, physics, and various other fields. A quadratic equation often gets solved using the quadratic formula, factoring, completing the square, or graphing.

• Structure: \(ax^2 + bx + c = 0\)
• Roots: \(\alpha\) and \(\beta\)

When dealing with more complex scenarios, understanding the behavior and relationships of its roots becomes crucial, which is where Vieta's formulas come in.

Vieta's Formulas

Vieta's formulas are powerful tools in algebra that relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation \(ax^2 + bx + c = 0\), they establish that:

• The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
• The product of the roots \(\alpha \beta = \frac{c}{a}\)

These relationships allow us to quickly deduce information about the roots without solving for them directly. In our specific example with \(375x^2 - 25x - 2 = 0\), using Vieta's formulas, we calculate:

• The sum of the roots to be \(\frac{1}{15}\)
• The product to be \(-\frac{2}{375}\)

Vieta's formulas are invaluable for deriving higher powers of the roots, leading into topics like recurrence relations where these powers play a central role.

Recurrence Relations

A recurrence relation is a way to define sequences with reference to previous terms. This method is central to understanding complex sequences in mathematics, such as those found in polynomials or series. For a sequence defined by a recurrence relation, each term is derived from one or more of its predecessors.In the context of quadratic equations and their roots, we use recurrence relations to relate terms like \(S_n = \alpha^n + \beta^n\). For our given problem, the sequence \(S_n\) follows:

• \(S_n = \frac{1}{15}S_{n-1} - \frac{2}{375}S_{n-2}\)

Such relations are particularly useful for evaluating compound expressions or series where closed-form solutions are not readily available. They simplify the calculations needed to understand how sequences evolve, especially when connected to other mathematical concepts, like geometric progressions.

Geometric Progression

A geometric progression (GP) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. If we think about powers of roots \(\alpha^n\) and \(\beta^n\), we can link these to geometric progressions because each power is derived by iteratively multiplying the root.In our problem, while analyzing the sequence \(S_n\), each \(\alpha^n\) and \(\beta^n\) can be visualized as terms in a geometric progression. This insight aids in grasping why these sequences rapidly approach zero as \(n\) increases; for roots with an absolute value less than one, their powers diminish quickly.The exploration of geometric progressions within recurrences provides a handy approach to estimate long-term behavior in sequences:

• Common Ratio: Typically less than 1, leading to decay
• Application: Helps justify rapid convergence to a limit

Understanding geometric progression equips you to tackle sequence analysis effectively, especially when combined with other mathematical constructs.