Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of `(2 4+1/(3 4))^n` is `sqrt(6)"":""1` .

Answer

N/a

Explanation

Total number of terms in the expansion of `(4sqrt(2)+(1)/(4sqrt(3)))^(n) " are " (n+1)`
`:.` 5th term from the beginning
`T_(5)=^(n)C_(4)(4sqrt(2))^(n-4)((1)/(4sqrt(3)))^(4)`
`=(|uln)/(|ul4|ul(n-4)).(4sqrt(2))^(n-4).((1)/(3))^(1/4xx4)`
`:. =(n(n-1)(n-2)(n-3))/(4.3.2.1).(4sqrt(2))^(n-4).((1)/(3)) .....(1)`
`:'` There are n+1 terms intiially.
`:.` 1st term from the end =(n+1) th term `=T_(n+1)`
2nd term from the end =nth term =`T_(n)`
3rd term from the end =(n-1)th term `-T_(n-2)`
4th term from the end =(n-2)th term `-T_(n-2)`
5th term from the end =(n-3)th term `=T_(n-3)`
Then 5th term from the end
`=T_(n-3)=^(n)C_(n-4)(4sqrt(2))^(n-(n-4)).((1)/(4sqrt(3)))^(n-4)`
`=(|uln)/(|ul(n-4).|ul4).2.((1)/(4sqrt(3)))^(n-4)`
`:. T_(n-3) =(n(n-1)(n-2)(n-3))/(4.3.2.1.).2. ((1)/(4sqrt(3)))^(n-4)....(2)`
Now the ratio of 5th term from beginning and the 5th term from the end
`(T_(5))/(T_(n-3))`
`((n(n-1)(n-2)(n-3))/(4.3.2.1)(4sqrt(2))^(n-4).(1/(3)))/((n(n-1)(n-2)(n-3))/(4.3.2.1).2.((1)/(4sqrt(3)))^(n-4))`
`((2)(n-4)/(4)(4sqrt(3))^(n-4))/(3.2)=((2)(n-4)/(4)(3)(n-4)/(4))/(6)`
`((2xx3)^((n-4)/(4)))/(6)`
`=(6)(n-4)/(4).6^(-1)=(6)"("((n-4))/(4)-1")"=(6)^(n/4-2)`
But it is given that the ratio of 5th term from beginning and the 5th from the end `=sqrt(6):1=(6)^(1/2)`
`:.6(n/4-2)=6^(1/2) rArr (n)/(4) -2=(1)/(2) " or " (n)/(4)=2+(1)/(2)`
`" or " n/4 =5/2 rArr n=10`
Therefore , n=10