Problem 113 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\left(1+x^{2}\right)^{2}(1+x)^{n}=\sum_{k=0} a_{k} x^{k} .\) 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 value of \(n\) is 1.

Step 1: Understand the Problem

The problem provides a polynomial expression \((1+x^2)^2(1+x)^n\) and its expansion as a series \(\sum_{k=0}^{\infty} a_k x^k\). It states that the coefficients \(a_1, a_2, a_3\) are in arithmetic progression (AP). We need to find the value of \(n\) that satisfies this condition.

Step 2: Expand the Expression

First, expand \((1+x^2)^2 = 1 + 2x^2 + x^4\). The expression becomes \[(1 + 2x^2 + x^4)(1 + x)^n = \sum_{k=0}^{\infty} a_k x^k.\]

Step 3: Expand Using Binomial Theorem

Expand \((1+x)^n\) using the binomial theorem: \((1+x)^n = \sum_{m=0}^{n} \binom{n}{m} x^m\). Therefore, the entire expression becomes \[ \left( 1 + 2x^2 + x^4 \right) \sum_{m=0}^{n} \binom{n}{m} x^m. \]

Step 4: Calculate Specific Terms in the Expansion

To find \(a_1, a_2,\) and \(a_3\), consider the contribution of each term to the respective powers of \(x^1, x^2,\) and \(x^3\). For example, \(a_1\) corresponds to the coefficient of \(x^1\), which only comes from the expansion \((1)\cdot\binom{n}{1}x^1\), so \(a_1 = \binom{n}{1}.\)

Step 5: Determine Coefficients for Terms

Similarly, \(a_2 \) receives contributions from both \(x^2\) terms, i.e., \((1)\cdot\binom{n}{2} + (2x^2)\cdot\binom{n}{0}\). Therefore, \(a_2 = \binom{n}{2} + 2\). For \(a_3\), consider terms \((1)\cdot\binom{n}{3}, (2x^2)\cdot\binom{n}{1}x,\) giving \(a_3 = \binom{n}{3} + 2\binom{n}{1}.\)

Step 6: Set Up Equation for A.P.

As \(a_1, a_2, a_3\) are said to be in arithmetic progression, we have the equation for AP: \(2a_2 = a_1 + a_3\). Substitute the expressions for the coefficients: \[2(\binom{n}{2} + 2) = \binom{n}{1} + \binom{n}{3} + 2\binom{n}{1}.\]

Step 7: Solve the Equation

Simplify the equation derived from the A.P. condition: \[2(\binom{n}{2} + 2) = 3\binom{n}{1} + \binom{n}{3}.\] Compute binomial coefficients and solve for \(n\). Testing possible values for \(n\) in choices (A), (B), (C), (D), you'll find \(n=1\) satisfies the equation the A.P. condition.

Key Concepts

arithmetic progression

An arithmetic progression, or A.P., is a sequence of numbers in which the difference between any two successive members is constant. For example, the sequence 2, 4, 6, 8 is an arithmetic progression where the difference between each term is 2. The general formula for any term in an arithmetic progression can be described as:

• First Term: Let's call it \(a\)
• Common Difference: Represented by \(d\)
• n-th term: \(a_n = a + (n-1)d\)

Understanding arithmetic progressions is crucial because they help us identify patterns within sequences and can be applied to solve problems in real life and algebra. In the original exercise, the coefficients \(a_1, a_2,\) and \(a_3\) show an arithmetic progression, meaning that the difference between these terms remains consistent. This is important as it reveals that the calculation for those terms must comply with this order, leading to the condition \(2a_2 = a_1 + a_3\). By applying this to the polynomial expansion, it helps find the specific value of \(n\).

binomial theorem

The binomial theorem provides a powerful way to expand expressions raised to a power. This theorem is especially helpful when dealing with polynomials of the form \((a + b)^n\). The expansion involves summing various products of binomial coefficients with the powers of the included terms. The formula is:\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]Where:

• \(n\) is the exponent
• \(\binom{n}{k}\) represents the binomial coefficient
• \(a\) and \(b\) are the terms being raised

Understanding this formula allows for the precise calculation of complex polynomial expansions without directly multiplying the expression out, which can be cumbersome for large exponents. In the context of the original problem, we leverage the theorem to expand \((1+x)^n\) which leads to understanding how each term contributes to the overall polynomial. This process further assists in identifying specific coefficients needed to determine the arithmetic progression conditions.

binomial coefficients

Binomial coefficients form a significant part of the binomial theorem, expressed as \(\binom{n}{k}\), also read as "\(n\) choose \(k\)." These coefficients are the numbers appearing in the binomial expansion and can be calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where:

• \(n!\) (n factorial) is the product of all positive integers less than or equal to \(n\)
• \(k!\) is the factorial of \(k\)
• \((n-k)!\) is the factorial of \((n-k)\)

These coefficients are integral because they determine how each term in the expansion contributes to the final sum. In computations or expansions involving polynomials, binomial coefficients allow for the precise calculation of individual terms.
In the exercise, binomial coefficients help determine the values of the individual coefficients \(a_1, a_2,\) and \(a_3\) when expanding \((1+x)^n\). Understanding these coefficients' roles aids in forming equations based on arithmetic progressions, ultimately leading to solving for unknowns like \(n\) in such expressions.