Question

Find the middle terms in the expansion of `(x^2 + 1/x)^7`

Answer 1

There are 8 terms in the expansion of `(x^2 + 1/x)^7`.

∴ middle terms are t₄ and t₅.

We know that, in the expansion of (a + b)ⁿ,

t_r+1 = ⁿC_r a^n–r b^r

Here a = x², b = `1/x`, n = 7

For t₄, r = 3 and for t₅, r = 4

∴ t₄ = `""^7"C"_3 (x^2)^(7 - 3) (1/x)^3`

= `(7 xx 6 xx 5)/(1 xx 2 xx 3) xx x^8 xx 1/x^3` = 35x⁵

and t₅ = `""^7"C"_4 (x^2)^(7 - 4) (1/x)^4`

= `(7 xx 6 xx 5 xx 4)/(1 xx 2 xx 3 xx 4) xx x^6 xx 1/x^4` = 35x²

 ∴ middle terms are 35x⁵ and 35x².