Problem 24 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{2 n} x^{2 n}\), then \(a_{0}+a_{3}+a_{6}+\ldots=\) (A) \(3^{n}\) (B) \(3^{n-1}\) (C) \(3^{n-2}\) (D) None of these

Step-by-Step Solution

Answer

The sum is \(3^{n-1}\), so option (B) is correct.

Step 1: Understanding the Expression

In this problem, we are given the expansion of \((1+x+x^2)^n\) as a polynomial: \(a_0 + a_1x + a_2x^2 + \ldots + a_{2n}x^{2n}\). We need to find the sum of the coefficients with indices that are multiples of 3, i.e., \(a_0 + a_3 + a_6 + \ldots\).

Step 2: Expression's Roots of Unity

To find the sum of coefficients at indices that are multiples of 3, we use the roots of unity filter. Consider evaluating the polynomial at the primitive cube roots of unity. The primitive cube roots are \(\omega = e^{2\pi i / 3}\) and \(\omega^2 = e^{4\pi i / 3}\).

Step 3: Using Roots of Unity

The key insight is that substituting \(x = 1\), \(x = \omega\), and \(x = \omega^2\) into the polynomial allows us to isolate the coefficients based on their modular index with respect to 3. Define \(S = a_0 + a_3 + a_6 + \ldots\).

Step 4: Evaluating at 1

Substituting \(x = 1\), the expression becomes \((1+1+1)^n = 3^n\). This is the sum of all coefficients: \(a_0 + a_1 + a_2 + \ldots + a_{2n} = 3^n\).

Step 5: Evaluating at Root \(\omega\)

Substituting \(x = \omega\), since \(1 + \omega + \omega^2 = 0\), the expression simplifies to \((0)^n = 0\). This is the sum \(a_0 + a_3\omega^3 + a_6\omega^6 + \ldots = 0\).

Step 6: Evaluating at Root \(\omega^2\)

Similarly, substituting \(x = \omega^2\), since \(1 + \omega^2 + \omega = 0\), we also obtain \(0\). Thus, \(a_0 + a_3\omega^3 + a_6\omega^6 + \ldots = 0\).

Step 7: Summing and Simplifying

From the evaluations, three variables are derived: \(A = a_0 + a_3 + a_6 + \ldots\), \(B = a_1 + a_4 + a_7 + \ldots\), \(C = a_2 + a_5 + a_8 + \ldots\). We have the equations: \(A + B + C = 3^n\), \(A + B\omega + C\omega^2 = 0\), \(A + B\omega^2 + C\omega = 0\). Solving these gives \(3A = 3^n\), thus \(A = 3^{n-1}\).

Step 8: Identifying the Correct Answer

Since \(A = 3^{n-1}\), the correct answer is option (B).

Key Concepts

Roots of Unity

The concept of roots of unity is fundamental in understanding symmetrical properties of polynomials, especially when solving problems involving repeating patterns. Roots of unity are complex numbers that, when raised to a certain integer power (usually the order), result in 1. Specifically, the { third roots of unity are used often because they are easy to visualize on the complex plane and find applications in solving cyclic patterns.}

• For cubic equations, the roots of unity are 1, \( \omega = e^{2\pi i / 3} \), and \(\ \omega^2 = e^{4\pi i / 3} \).
• The number 1 is known as the trivial root, while \( \omega \) and \(\ \omega^2 \) are the non-trivial roots.
• These roots satisfy the equation \( \omega^3 = 1 \) and \( \omega + \omega^2 + 1 = 0 \).

By evaluating a given polynomial expression at these roots, you can effectively filter coefficients that repeat periodically, thereby isolating and summing specific terms as needed. This filtering is crucial when coefficients follow a repeating cycle, such as every third term, which aligns perfectly with the properties of the cube roots of unity.

Polynomial Expansion

A polynomial expansion involves expressing a polynomial as a sum of terms composed of a base raised to an exponent and often includes coefficients. The binomial theorem allows us to expand expressions like \(\ (1+x+x^2)^n \) efficiently.

• The basic idea is to expand the given expression in powers, often using known identities or formulas.
• In this exercise, the polynomial is expanded to form terms like \(\ a_0 + a_1 x + a_2 x^2 + \ldots + a_{2n} x^{2n} \), where \(\ n \) determines the depth of the expansion.
• Each coefficient, \(\ a_i \), relates to a particular combination of the base terms used in the expansion.}

Expanding polynomials is crucial because it allows us to analyze and manipulate expressions easily, obtaining insights into their structure. This exercise specifically seeks to sum certain coefficients, making expansion methods handy for simplification before applying more advanced theorems or filters, such as those based on roots of unity.

Coefficient Summation

Coefficient summation is a technique used to extract particular terms from a polynomial expression based on certain conditions. In this problem, the goal is to sum coefficients whose indices are multiples of three.

• By utilizing roots of unity, specifically the third roots, one can succinctly determine the sum of coefficients appearing at these regular intervals.
• Substituting values for \(x = 1, \omega, \omega^2 \) allows a strategic cancellation of terms, demonstrating symmetry and achieving clarity in sums by isolating coefficients meeting our specified condition.
• The derived equations assist in solving unknowns as coefficients align along the symmetry grid set by these chosen roots.

For this exercise, setting and solving equations derived from these substitutions provides that the sum of every third coefficient is \(\ 3^{n-1} \). This solution demonstrates the elegance and utility of these mathematical concepts in parsing seemingly complex polynomial relationships into simpler comprehendible results.