Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of

$$(\sqrt{2} +\dfrac{1}{\sqrt{3} })^n$$ is $$\sqrt{6}: 1$$

Asked by Pragya Singh | 1 year ago |  153

##### Solution :-

In the expansion,

$$(a+b)^n=\;^nC_0a^n+^nC_1a^{n-1}+^nC_2a^{n-2}b^2$$

$$+......+^nC_{n-1}+^nC_nb^n$$

Fifth term from the beginning $$^nC_4a^{n-4}b^4$$

Fifth term from the end = $$^nC_4a^4b^{4-4}$$

Therefore, it is evident that in the expansion of

$$(\sqrt{2} +\dfrac{1}{\sqrt{3} })^n$$ are fifth term from the beginning is

$$^nC_4(\sqrt{2})^{n-4}(\dfrac{1}{\sqrt{2}})^4$$ and the fifth term from the end is

$$^nC_{n-4}(\sqrt{2})^4(\dfrac{1}{\sqrt{3}})^{n-4}$$

$$^nC_4(\sqrt{2})^{n-4}(\dfrac{1}{\sqrt{3}})^4$$

$$^nC_4\dfrac{(\sqrt{2})^n}{(\sqrt{2})^n}\times \dfrac{1}{3}$$

$$=\dfrac{n!}{6\times 4!(n-4)!}(\sqrt{2})^n$$..........(1)

$$^nC_{n-4}(\sqrt{2})^n(\dfrac{1}{\sqrt{2}})^{n-4}$$

$$^nC_{n-4}\dfrac{(\sqrt{3})^4}{(\sqrt{3})^n}$$

$$=^nC_{n-4}\times 2\times \dfrac{3}{(\sqrt{3})^n}$$

$$\dfrac{6!}{(n-4)!4!}\times \dfrac{1}{(\sqrt{3})^n}$$ ...........(2)

It is given that the ratio of the fifth term from the beginning to the fifth term from the end is $$\sqrt{ 6} :1.$$ Therefore, from (1) and (2), we obtain

$$=\dfrac{n!}{6\times 4!(n-4)!}(\sqrt{2})^n:$$

=  $$\dfrac{6!}{(n-4)!4!}\times \dfrac{1}{(\sqrt{3})^n}=\sqrt{ 6} :1$$

$$=\dfrac{(\sqrt{2})^n}{6}:\dfrac{6}{(\sqrt{3})^n}=\sqrt{ 6} :1$$

$$(\sqrt{2})^n=36 \sqrt{ 6}$$

$$6^\dfrac{n}{4}=6^\dfrac{5}{2}$$

$$\dfrac{n}{4}=\dfrac{5}{2}$$

$$n=4\times \dfrac{5}{2}=10$$

Thus, the value of n is 10.

Answered by Abhisek | 1 year ago

### Related Questions

#### Find the term independent of x in the expansion of (3/2 x2 – 1/3x)9

Find the term independent of x in the expansion of $$(\dfrac{3}{2x^2} – \dfrac{1}{3x})^9$$

#### Find the middle term in the expansion of (x – 1/x)2n+1

Find the middle term in the expansion of $$(x-\dfrac{ 1}{x})^{2n+1}$$

#### Find the middle term in the expansion of (1 + 3x + 3x2 + x3)2n

Find the middle term in the expansion of (1 + 3x + 3x2 + x3)2n

Find the middle term in the expansion of $$(\dfrac{x}{a} – \dfrac{a}{x})^{10}$$