If \(p\) is a real number and if the middle term in the expansion of \(\left(\dfrac{p}{2}+2\right)^8\) is 1120, find \(p\).
\(p=\pm 2\)
Show that the middle term in the expansion of \(\left(x-\dfrac{1}{x}\right)^{2n}\) is \(\dfrac{1\times3\times5\times\dots\times(2n-1)}{n!}\times(-2)^n\).
Find \(n\) in the binomial \(\left(\sqrt[3]{2}+\dfrac{1}{\sqrt[3]{3}}\right)^n\) if the ratio of the 7th term from the beginning to the 7th term from the end is \(\dfrac{1}{6}\).
\(n=9\)
In the expansion of \((x+a)^n\) if the sum of odd terms is denoted by \(O\) and the sum of even terms by \(E\), prove that:
(i) \(O^2-E^2=(x^2-a^2)^n\)
(ii) \(4OE=(x+a)^{2n}-(x-a)^{2n}\)
If \(x^p\) occurs in the expansion of \(\left(x^2+\dfrac{1}{x}\right)^{2n}\), prove that its coefficient is \(\dbinom{2n}{\dfrac{2n+p}{3}}\) (for integral \(\dfrac{2n+p}{3}\)).
Find the term independent of \(x\) in the expansion of \(\left(\dfrac{3}{2}x^2-\dfrac{1}{3x}\right)^9\).