Find the middle term in the expansion of [x-1/x]10 - Maths - Sequences and Series

Question

Find the middle term in the expansion of
[x-1/x]10

Lalit Mehra · Accepted Answer

Dear Student!

The given expansion is $(x - \frac{1}{x})^{10}$ .

General term of the given expansion, $t_{r + 1} = (- 1)^{r}^{10} C_{r} x^{10 - r} (\frac{1}{x})^{r}$

Here n = 10, which is even.

So, $(\frac{10}{2} + 1)$ th term i.e., 6^th term is the middle terms of the given expansion.

Putting r = 5, we get

$t_{5 + 1} = (- 1)^{5}^{10} C_{5} x^{10 - 5} (\frac{1}{x})^{5}$

$∴ t_{6} = -^{10} C_{5} x^{5} \times \frac{1}{x^{5}} = -^{10} C_{5} = - 252$

Thus, the middle term in the given expansion is –252.

Cheers!