Question

coefficient of x^39 in the expansion of [(1/x^2) + x^4)]^18

Answer 1

Use the general term

term(n+1) = C(18,n) (1/x^2)^(18-n) (x^4)^n

= C(18,n) (x^(-36 +2n) ) (x^4n)
= c(18,n) x^(2n - 36)

ahhh, trick question,
to have a term with x^39 , 2n-36 = 39
there is no whole number solution,
<b>so there is no term with x^39</b>

e.g. --- take the first few terms
[(1/x^2) + x^4)]^18
= (1/x^2)^18 + 18(1/x^2)^17 (x^4) + 153 (1/x^2)^16 (x^4)^2 + ..
= x^-36 + 18x^-30 + 153x^-24 + ..
the exponent go
-36 -30 -24 .. missing the 39