Question

The (m + 1)ᵗʰ term of $\left(\frac{x}{y}+\frac{y}{x}\right)^{2m+1}$ is:

• A. independent of x
• B. a constant
• C. depends on the ratio x/y and m
• D. none of these

Answer 1

Answer: (C)

(C) depends on the ratio x/y and m

Answer 2

The general term of the binomial expansion $(a+b)^n$ is given by $T_{r+1} = ^nC_r a^{n-r} b^r$. In this case, $a = \frac{x}{y}$, $b = \frac{y}{x}$, and $n = 2m+1$. We need the $(m+1)^{th}$ term, so $r+1 = m+1 \implies r=m$. The $(m+1)^{th}$ term is $T_{m+1} = ^{2m+1}C_m \left(\frac{x}{y}\right)^{(2m+1)-m} \left(\frac{y}{x}\right)^m = ^{2m+1}C_m \left(\frac{x}{y}\right)^{m+1} \left(\frac{y}{x}\right)^m = ^{2m+1}C_m \frac{x^{m+1}}{y^{m+1}} \frac{y^m}{x^m} = ^{2m+1}C_m x^{m+1-m} y^{m-(m+1)} = ^{2m+1}C_m x^1 y^{-1} = ^{2m+1}C_m \frac{x}{y}$. This term depends on the ratio $\frac{x}{y}$ and on $m$ (through the binomial coefficient $^{2m+1}C_m$).