+88 (a) For what positive integers \(n\) does \(\left(x^2+\frac{1}{x}\right)^n\) have a nonzero constant term?
(b) For the values of \(n\) that you found in part (a), what is that constant term? (You can leave your answer in the form of a combination.)
I need an INCREDIBLY thorough explanation please!!!!!
THANKS FOR ALL HELP
Im so cunfused i deseratly need help
+118667 a) n can be any multiple of 3.
You have a habit of telling people that their answers are wrong.
So if you are going to say this then you might as well say it now before I explain my answer.
+129849 To see Melody's answer
Note that......when n = 1
(x^2 + 1/x)^(3*1) = (x^2 + 1/x)^3 will have the term C(3,2)(x^2)^1 * (1/x)^2 = 3x^2(1/x^2)
Constant term = 3
When n = 2.....
(x^2 + 1/x)^(3*2) = (x^2 + 1/x)^6 will have the term C(6, 4)(x^2)^2 * (1/x)^4 = 15 (x^4)(1/x^4)
Constant term = 15
When n = 3....
(x^2 + 1/x)(3 * 3) will have the term C(9, 6) (x^2)^3 *(1/x)^6 = 84 (x^6)(1/x^6)
Constant term = 84
So.....it appears the the pattern for the constant term will be
C(3n, 2n) * (x^2)^n * (1/x)^(2n) where n is a positive integer
+118667 ok here is my logic.
\((x^2+\frac{1}{x})^n\)
let a be an integer such that \(0\le a \le n\)
the a'th term will be
\(\begin{pmatrix}n\\a\end{pmatrix}[x^2]^a\;\left[ \frac{1}{x} \right]^{n-a}\\ \begin{pmatrix}n\\a\end{pmatrix}x^{2a}\;\times x ^{a-n}\\ \begin{pmatrix}n\\a\end{pmatrix}x^{3a-n}\\\ \)
this will be a constant term whenever
\(3a-n=0\\n=3a\)
Since a is an integer, the expression will have a constant term if n is a multiple of 3
The value of the constant term is already included in my answer.
