Problem 13 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of distinct terms in the expansion of \(\left(x^{3}+1+\frac{1}{x^{3}}\right)^{n} ; x \in R^{+}\)and \(n \in N\) is (A) \(2 n\) (B) \(3 n\) (C) \(2 n+1\) (C) \(3 n+1\)

Step-by-Step Solution

Answer

The number of distinct terms is \(2n + 1\).

Step 1: Identify the Terms

In the expansion of \((x^3 + 1 + \frac{1}{x^3})^n\), each term is of the form \(x^{3a} \cdot 1^b \cdot \left(\frac{1}{x^3}\right)^c\), where \(a + b + c = n\). Thus, the power of \(x\) in each term is \(3a - 3c\).

Step 2: Expression for Power of x

The expression for the power of \(x\) in any term is \(3(a - c)\). This implies the power of \(x\) is a multiple of 3. The minimum value of \(a - c\) is \(-n\) and the maximum value is \(n\). Hence, the power ranges from \(-3n\) to \(3n\).

Step 3: Counting Distinct Powers

The powers of \(x\) range from \(-3n\) to \(3n\), inclusive. These are given by \(-3n, -3n+3, -3n+6, ..., 0, ..., 3n-6, 3n-3, 3n\).

Step 4: Determine Number of Distinct Powers

The series \(-3n, -3n+3, ..., 3n\) is an arithmetic sequence with a common difference of 3. The number of terms \(T\) in this sequence can be calculated by using the formula for the number of terms in an arithmetic sequence: \(T = \frac{\text{last term} - \text{first term}}{\text{difference}} + 1\).

Step 5: Calculate Number of Distinct Terms

Plug in the values: \(T = \frac{3n - (-3n)}{3} + 1 = \frac{6n}{3} + 1 = 2n + 1\). There are \(2n + 1\) distinct powers of \(x\) in the expansion.

Key Concepts

Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference. In the context of the problem, the powers of \(x\) form an arithmetic sequence.

• The powers are presented as: \(-3n, -3n+3, -3n+6, \ldots, 3n-6, 3n-3, 3n\).
• The common difference in this sequence is 3.

To determine the number of terms in this sequence, we use the formula:\[T = \frac{\text{last term} - \text{first term}}{\text{difference}} + 1\]By applying this to our sequence:\[T = \frac{3n - (-3n)}{3} + 1 = \frac{6n}{3} + 1 = 2n + 1\]This shows there are \(2n + 1\) terms in this arithmetic sequence.

Distinct Terms

The concept of distinct terms in a binomial expansion is critical, as it helps in understanding the simplicity or complexity of the expansion's structure. For this exercise, each term in the expansion is identified by its power of \(x\), which could range from a minimum to a maximum value due to the nature of the binomial expansion.- Each term in the expansion form is \(x^{3a - 3c}\).- The key point here is to find all different values that the power of \(x\) can take.The terms considered distinct when their powers of \(x\) vary. From the arithmetic sequence earlier and the formula, we deduced that the number of distinct terms is \(2n + 1\). This is crucial as it indicates how many unique combinations of terms contribute to the overall expression.

Power of a Term

In binomial expansions, the power of each term significantly influences the number of distinct terms and the pattern. Here, every term resulting from the expansion is represented by a specific power of \(x\), calculated through the formula \(3(a-c)\).- Since \(a + b + c = n\), we play with these combinations by varying \(a\), \(b\), and \(c\) while maintaining the sum equal to \(n\).- The minimum power of \(x\) occurs when \(a-c = -n\), leading to the power \(-3n\).- The maximum power of \(x\) is reached when \(a-c = n\), making the power \(3n\).This range of powers translates directly into the distinct terms present in the sequence, from \(-3n\) to \(3n\). The understanding of how this power changes is essential in counting distinct terms, which collectively form the comprehensive binomial expansion.