Question

Find the term independent of x in the expansion of

`(2x^2 + 1/x)^12`

Answer 1

`(2x^2 + 1/x)^12` Compare with the (x + a)ⁿ.

Here x is 2x², a is `1/x`, n = 12.

Let the independent term of x occurs in the general term.

t_r+1 = nC_r x^n-r a^r

`"t"_(r+1) = 12"C"_r (2x^2)^(12-r) (1/x)^r = 12"C"_r  2^(12-r) x^(2(12-r)) x^-r`

= 12C_r 2^12-r x^24-2r x^-r

`= 12"C"_r  12^(12-r) x^(24-3r)`

Independent term occurs only when x power is zero

24 – 3r = 0

24 = 3r

r = 8

Put r = 8 in (1) we get the independent term as

= 12C₈ 2^12-8 x⁰

= 12C₄ × 2⁴ × 1

= 7920