(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.)
+26367 (a)
For what positive integers \(n\) does \(\displaystyle \left(x^2+\frac{1}{x}\right)^n\) have a nonzero constant term?
\(\begin{array}{|rcll|} \hline && \left(x^2+\dfrac{1}{x}\right)^n \\\\ &=& \left(\dfrac{x^3+1}{x}\right)^n \\\\ &=& \dfrac{1}{x^n} \cdot \left(1+ x^3 \right)^n \\\\ &=& \dfrac{1}{x^n} \cdot \left(\dbinom{n}{0} + \dbinom{n}{1}\left(x^3 \right)^1 + \dbinom{n}{2}\left(x^3 \right)^2 + \dbinom{n}{3}\left(x^3 \right)^3 + \ldots + \dbinom{n}{n}\left(x^3 \right)^n \right) \\\\ &=& \dfrac{1}{x^n} \cdot \sum \limits_{i=0}^{n} \dbinom{n}{i} \left(x^3 \right)^i \\ \hline \end{array}\)
\(\text{We have a nonzero constant, if $\\ \dfrac{1}{x^n} \cdot \dbinom{n}{i} \left(x^3 \right)^i = \dbinom{n}{i}$, $\ $ } \text{so $\left(x^3 \right)^i = x^{3i} = x^n $,$\ $ } \text{so $ \mathbf{3i = n}$ } \text{and $ i \in N $ }\)
The positive integers \(n\), which are 3 or multiples of 3 have a nonzero constant term.
\(n=3,6,9,12,\ldots.\)
(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.)
\(\begin{array}{|r|r|l|} \hline i & n=3i & \dbinom{n}{i} \\ \hline 1 & 3 & \dbinom{3}{1} = 3 \\ \hline 2 & 6 & \dbinom{6}{2} = 15 \\ \hline 3 & 9 & \dbinom{9}{3} = 84 \\ \hline 4 & 12 &\dbinom{12}{4} = 495 \\ \hline \ldots & \ldots & \ldots \\ \hline \end{array} \)
