Problem 113 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\left(1+x^{2}\right)^{2}(1+x)^{n}=\sum_{k=0}^{n+4} a_{k} x^{k}\). If \(a_{1}, a_{2}, a_{3}\), are in A.P., then \(n\) is equal to (A) 1 (B) 2 (C) 3 (D) 4

Step-by-Step Solution

Answer

The solution shows that \(n = 3\).

Step 1: Expand the expression

We are given the expression \((1+x^2)^2(1+x)^n\). First, expand \((1+x^2)^2\) as it simplifies to \(1 + 2x^2 + x^4\).

Step 2: Expand the combined product

Next, expand \((1+2x^2+x^4)(1+x)^n\). Use the binomial expansion formula: \((1+x)^n = \sum_{j=0}^{n} \binom{n}{j} x^j\). Substitute this into the expression to obtain: \(1 + 2x^2 + x^4 = (1+x)^n + 2x^2(1+x)^n + x^4(1+x)^n\).

Step 3: Find general coefficients

For each term in the expanded expression, calculate the coefficients of terms up to \(x^4\), as these will directly influence the coefficients \(a_1\), \(a_2\), and \(a_3\).

Step 4: Calculate specific coefficients

Calculate the coefficients \(a_1\), \(a_2\), and \(a_3\) for each polynomial term:- For \(a_1\), only the linear term from \((1+x)^n\) contributes, giving \(a_1 = n\).- For \(a_2\), the terms are \(\binom{n}{2}\) from \((1+x)^n\) and \(2\) from \(2x^2\), so \(a_2 = \binom{n}{2} + 2\).- Solve similarly for \(a_3\).

Step 5: Use A.P. relation

Given that \(a_1, a_2, a_3\) are in arithmetic progression (A.P.), fulfill the condition: \(2a_2 = a_1 + a_3\). Substitute and simplify to solve for \(n\).

Step 6: Verify and conclude

After substitution and simplification using the A.P. condition, solve for integer values of \(n\) and determine which value satisfies the equation.

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the "common difference."
For example, in the sequence 2, 4, 6, 8, the common difference is 2. Arithmetic progressions can be written in the form:

• First term: \(a\)
• Common difference: \(d\)
• General term: \(a_n = a + (n-1)d\)

In the context of the problem, the given coefficients \(a_1, a_2, a_3\) are part of an arithmetic progression. This means that the relationship \(2a_2 = a_1 + a_3\) holds true, where \(a_2\) is the average of the other two terms.
This concept helps us solve for the variable \(n\) by setting up an equation based on the formulas of the progression, ensuring all entries follow this simple relationship.

Binomial Expansion

The binomial expansion is a powerful algebraic tool used to expand expressions of the form \((a+b)^n\). This is articulated in a formula that employs binomial coefficients:

• \((a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\)

In the given problem, we expand \((1+x)^n\) using this formula, where \(a=1\) and \(b=x\). This helps us find the coefficients of various terms in the expansion.
The binomial coefficient, represented as \(\binom{n}{k}\), plays a crucial role in determining how the powers of \(x\) appear in the expanded expression. Understanding how to calculate and apply these coefficients allows us to find specific terms and their coefficients quickly. Applying the expansion, we calculate each power term for up to \(x^4\) to explore the desired sequence. This expansion allows us to navigate complex polynomial expressions by breaking down each term systematically.

Polynomial Coefficients

Polynomial coefficients are the numerical factors connected with the terms of a polynomial. In any polynomial of the form \(ax^n + bx^{n-1} + \ldots + k\), \(a, b,\ldots, k\) are the coefficients.
Within the given problem, coefficients \(a_1, a_2, a_3\) emerge in relation to the polynomial

• \((1+x^2)^2(1+x)^n = \sum_{k=0}^{n+4} a_k x^k\)

In this sense, \(a_k\) denotes the coefficient for \(x^k\). Understanding how to extract these based on expansion mechanisms is critical. Step by step, each component of the expression contributes to the final coefficients through multiplication and algebraic simplification.
To solve the problem, one must determine \(a_1, a_2, a_3\):

• \(a_1\) is derived from the linear term coefficient.
• \(a_2\) comes from quadratic contributions.
• \(a_3\) requires considering terms involved in higher powers.

Accurate calculation of these coefficients ensures the solution respects the given arithmetic progression, guiding us to the correct integer \(n\).