Question

Find the Co-efficient of x¹¹ in the expansion of `(x + 2/x^2)^17`

Answer 1

In `(x + 2/x^2)^17`, n = 17, x = x, a = `2/x^2`

∴ The general terms is

`"t"_(r + 1) = n"C"_r  x^(n - r) a^r`

`= 17C_r  x^(17-r) (2/x^2)^r`

`= 17C_r  x^(17-r) * 2^r/(x^(2r))`

`= 17C_r * 2^r  x^(17 - 3r)`    ....(1)

To get the co-efficient of x¹¹,

⇒ 17 - 3r = 11

⇒ 17 - 11 = 3r

⇒ 3r = 6

⇒ r = 2

Put r = 2 in (1) we get,

`"t"_3 = 17"C"_2  2^2 x^(17-3(2))`

= 17C₂(4)x¹¹

`= (17 xx 16)/(2xx1) xx 4 * x^11`

= 544 x¹¹

∴ Co-efficient of x¹¹ is 544.