Question

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ($\frac{\sqrt{24}+1}{\sqrt{34}}$)n, in the increasing powers of 134 be 64:1.If the sixth term from the beginning is $\frac{\alpha}{\sqrt{34}}$, then α is equal to ___________.

Answer 1

Fifth term from the beginning
=nC4($\frac{21}{4}$)n−4(3−$\frac{1}{4}$)4
Fifth term from end = (n – 5 + 1)th term from begin
=nCn−4($\frac{21}{4}$)3($\frac{3-1}{4}$)n−4
Given
nC42n−44⋅3−$^{\frac{1}{n}}$Cn−$\frac{324}{4}$⋅3−($\frac{n-4}{4}$)=$\frac{64}{4}$
⇒$\frac{6n-8}{4}$=$\frac{61}{4}$
⇒$\frac{n-8}{4}$=$\frac{1}{4}$ ⇒n=9
T6=T5+1=9C5($\frac{21}{4}$)4(3−$\frac{1}{4}$)5
=9C5⋅$\frac{231}{4.3}$=$\frac{8431}{4}$=$\frac{\alpha31}{4}$
⇒ α = 84.