Problem 83 - A Complete Resource Book in Mathematics for JEE Main

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 answer is (B) \(3^{n-1}\).

Step 1: Recognize the Pattern

The exercise requires finding the sum of coefficients of terms where the powers of x are multiples of 3. This pattern is expressed as \(a_0 + a_3 + a_6 + \ldots\).

Step 2: Use the Roots of Unity Filter

To pick out coefficients of terms where the exponents are multiples of 3, substitute \(\omega = e^{2\pi i / 3}\), a cube root of unity. Specifically, we will use the identity \(1 + \omega + \omega^2 = 0\).

Step 3: Apply the Substitution in the Polynomial

Substitute \(x = 1\), \(x = \omega\), and \(x = \omega^2\) into \((1+x+x^2)^n\):1. When \(x = 1\), the result is \((3)^n\).2. When \(x = \omega\), the result is \((1 + \omega + \omega^2)^n = 0^n = 0\).3. When \(x = \omega^2\), the result is \((1 + \omega^2 + \omega)^n = 0^n = 0\).

Step 4: Calculate the Desired Sum

The sum \(a_0 + a_3 + a_6 + \ldots\) is given by \(\frac{(1)^n + (\omega)^n + (\omega^2)^n}{3}\).Since \((\omega)^n = 0\) and \((\omega^2)^n = 0\), the expression simplifies to \(\frac{3^n}{3} = 3^{n-1}\).

Key Concepts

Roots of Unity

Roots of unity are fascinating mathematical concepts that play a key role in various fields, including number theory, algebra, and geometry. In simpler terms, a root of unity is a complex number that, when raised to a certain power, yields 1. For example, the cube roots of unity are numbers that, when cubed, produce 1.
The most well-known cube roots of unity are 1, \(\omega\), and \(\omega^2\). Here, \(\omega\) is defined as \(e^{2\pi i / 3}\), a complex number situated on the unit circle in the complex plane. Its corresponding values satisfy the simple but mighty equation: \(1 + \omega + \omega^2 = 0\). This identity is particularly important as it helps simplify many equations involving roots of unity.
By exploiting this identity in the polynomial expansion problem, we effectively use the roots of unity as a filter to isolate certain terms or coefficients. In practice, this approach makes use of the substitution of \(x\) by \(\omega\) and \(\omega^2\) to filter out coefficients associated with specific powers, like multiples of 3 in our problem's equation.

Polynomial Expansion

Polynomial expansion is a fundamental concept in algebra, allowing us to express a polynomial as a sum of terms with coefficients. In the exercise at hand, the polynomial \( (1 + x + x^2)^n \) is expanded to reveal its constituent terms and coefficients.
Understanding polynomial expansion is crucial because it reveals how different terms contribute to the expanded result. Each term's contribution depends on its coefficient, which is composed of specific combinations of operation outcomes applied repeatedly. We use expansion techniques such as the binomial theorem to manage complex expansions and to calculate specific coefficients like \(a_0, a_3, a_6,...\) in our case. Here, recognizing where each coefficient stands within the polynomial is simplified by the methodical use of algebraic identities, such as those derived from roots of unity.
Knowing how to apply polynomial expansion not only allows us to solve equations like the one in the exercise but also helps in comprehending polynomial expressions in algebra problems, calculus series, signal processing, and other areas. It is a full circle of multiplication, division (factorization), addition, and the exploration of the pattern each polynomial holds.

Combinatorics

Combinatorics, the branch of mathematics concerned with counting, is closely linked to problems of arrangement and selection, which are common in polynomial expressions. Although the current exercise doesn't deeply delve into combinatorics for its solution, the methodology still subtly incorporates combinatorial thinking.
When we consider the polynomial expansion, for each \(a_i\), representing the coefficient of \(x^i\), a combinatorial scenario can be bridged, where sequences of operations yield the specific terms. These coefficients can be a result of various combinations of rooted expansions based on principles of symmetry and systematic grouping of terms. Using properties like the roots of unity, we indirectly apply combinatorial concepts by considering all formations of the terms and selectively observing those aligning with criteria like multiples of 3. In a broader view, this aligns with the combinatorial approach of organizing elements into structured patterns, fulfilling set conditions.
Thus, understanding combinatorics helps to unravel not just numerical expansions but also interpret the structured harmony among the resulting expressions and coefficients—a subtle yet powerful tool in mathematics.