InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Show that the middle term in the expansion of `(1+x)^(2n)i s((1. 3. 5 (2n-1)))/(n !)2^n x^n ,w h e r en`is a positive integer. |
|
Answer» Clearly, the number of terms in the expansion of `(1+x)^(2n)" is "(2n+1).` `:. " middle term " =((2n)/2+1)" th term "= (n+1)" th term "= T _(n+1).` Now, `T_(n+1) = .^(2n) C _(n)* x^(n) = ((2n)!)/((n!) xx (n !))* x ^(n) ` `=(1* 2* 3 * 4 ... ( 2n - 2) (2n-1) (2n))/((n!) xx ( n !)) * x^(n)` `=( [ 1 * 3 * 5 ... ( 2n - 3) ( 2n-1)] xx [2* 4* 6 ... ( 2n - 2) (2n) ])/((n!) (n!)) * x^(n) ` `=([1 * 3* 5 ... ( 2n-1)] xx 2 ^(n) xx [ 1* 2 * 3 ... ( n-1) *n])/((n!) xx (n!))* x^(n)` `= ([ 1* 3* 5 ... ( 2n-1)] xx 2^(n) xx x^(n))/((n!))*` Hence , the middle term in the given expansion is `(1* 3* 5 ... ( 2n-1) xx 2^(n) xx x^(n))/((n!))*` |
|