Question

Find the middle terms in the expansion of i) $\left(\frac{x}{y}+\frac{y}{x}\right)^{12}$

Answer 1

924 and 792*(y^2/x^2)

Answer 2

The general term of the expansion

$$\left(\frac{x}{y}+\frac{y}{x}\right)^{12}$$

is

$$T_{r+1}=\binom{12}{r}\left(\frac{x}{y}\right)^{12-r}\left(\frac{y}{x}\right)^r = \binom{12}{r}x^{12-2r}y^{2r-12}.$$

Since there are $13$ terms, the two middle terms are for $r=6$ and $r=7$.

• For $r=6$:

  $$T_7=\binom{12}{6}x^{12-12}y^{12-12}=\binom{12}{6}=924.$$

• For $r=7$:

  $$T_8=\binom{12}{7}x^{12-14}y^{14-12}=\binom{12}{7}\frac{y^2}{x^2}=792\,\frac{y^2}{x^2}.$$

The middle terms are:

$$924 \quad \text{and} \quad 792\,\frac{y^2}{x^2}.$$