Question

In \((x+y)^{n}-(x-y)^{n}\) if the number of terms is 5, then find \(n\). (1) 6 (2) 5 (3) 10 (4) \(\overline{9}\)

Short answer

Answer: (4) 9

Step-by-step solution

Understand the Binomial Theorem

The binomial theorem states that for any positive integer \(n\), the expansion of \((x+y)^{n}\) is given by $$(x+y)^{n} = \sum_{r=0}^n \binom{n}{r}x^{n-r}y^{r}$$ where, \(\binom{n}{r}=\frac{n!}{(n-r)!r!}\) is the binomial coefficient.

Expand the given expression using the binomial theorem

Now, let's expand the given expression (\((x+y)^{n}-(x-y)^{n}\)) using the binomial theorem: $$ (x+y)^{n}-(x-y)^{n} = \sum_{r=0}^n \binom{n}{r}x^{n-r}(y^r-(-y)^r) $$

Simplify the expanded expression

Now, we can simplify the expanded expression using the property \((-1)^r=-1\) if \(r\) is odd, and \((-1)^r=1\) if \(r\) is even: $$ \sum_{r=0}^n \binom{n}{r}x^{n-r}(y^r-(-y)^r) = \sum_{\substack{r=0 \\ 2|r}}^n 2 \binom{n}{r}x^{n-r}y^r $$ In this simplified expression, we only consider the even values of \(r\), so we will have \((n+1)/2\) terms (in case n is odd) and \(n/2 + 1\) terms (in case n is even).

Determine the value of n that gives 5 terms

Since the number of terms is 5, if n is odd then the equation \((n+1)/2 = 5\) should be satisfied, if n is even then the equation \(n/2 + 1 = 5\) should be satisfied. For odd n, \((n+1)/2 = 5\): $$ n+1 = 10 \\ n = 9 $$ For even n, \(n/2 + 1 = 5\): $$ n/2 = 4 \\ n = 8 $$ Neither of the options (1,2,3,4) matches these values of n (9 or 8). So, we can safely assume this is a mistake in the problem (we added \(\overline{9}\)). Thus, the answer is \((4)\ \overline{9}\).

Key concepts

Binomial Expansion

The binomial theorem is a powerful tool for expanding expressions of the form \((x+y)^{n}\). It allows us to express the power as a series of terms, each involving a binomial coefficient. The general formula is:

• \((x+y)^{n} = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^{r}\)
• Each term in the expansion corresponds to a coefficient \( \binom{n}{r} \).

This formula helps simplify complex polynomials into manageable parts. It's especially useful for solving problems involving large powers.

Coefficients

In the context of the binomial expansion, coefficients are represented by the notation \(\binom{n}{r}\), known as binomial coefficients. These are calculated using the formula:

• \(\binom{n}{r} = \frac{n!}{(n-r)!r!}\)
• These coefficients determine the weight or value of each term in the expansion.

Understanding coefficients is key to determining how each part of the polynomial contributes to the final expression. They are fundamental in algebra and combinatorics.

Combinations

Combinations play a crucial role in calculating binomial coefficients. A combination \(\binom{n}{r}\) represents the number of ways to choose \(r\) items from \(n\) without regard to order. It's given by:

• \(\binom{n}{r} = \frac{n!}{(n-r)!r!}\)
• This formula helps in various combinatorial problems and is foundational in probability and statistics.

Identifying the correct combination helps in solving equations involving binomial expansions efficiently.

Mathematical Expression Simplification

Simplifying mathematical expressions is essential for clarity and efficiency. In this problem, we simplified \( (x+y)^{n}-(x-y)^{n} \) using properties of numbers and the binomial theorem:

• Recognize patterns to group and reduce terms based on parity (odd/even).
• For this expression: subtract terms to simplify by focusing on even values of \(r\), resulting in a reduced number of terms.

Simplification reduces complexity and helps solve problems faster by minimizing potential errors and computational work.