The greatest coefficient in the expansion of `(1+x)^(2n)` is :

Options

A. `((n-1)/(n),(n)/(n-1))`
B. `((n)/(n+1),(n+1)/(n))`
C. `((n)/(n+2),(n+2)/(n))`
D. none of these

Answer

b

Explanation

Let `(r +1)^(th)` term be the greatest term in the
expansion of `(1 +x)^(2n)`. Then ,
`T_(r+1) = ""^(2n)C_(r) x^(2n-r)`
The coefficient of `T_(r+1) is ""^(2n)C_(r)` Clearly, `""^(2n)C_(r)` willl be greatest ,if
` r = (2n)/(2) = n`
Thus , `T_(r+1) = ""^(2n)C_(n) x^(n)` has the greatest coefficient.
Now,
Thus `T_(n+1)` wil be the greatest term , if
`T_(n) lt T_(n+1) gtT_(n+2)`
`rArr T_(n+1) gt T_(n) and T_(n+2) lt T_(n+1)`
`rArr (T_(n+1))/(T_(n)) gt 1 and (T_(n+2))/(T_(n +1)) lt `
`rArr (""^(2n)C_(n) x^(n))/(""^(2n)C_(n-1) x^(n-1)) gt 1 and (""^(2n)C_(n+1) x^(n+1))/(""^(2n)C_(n) x^(n))lt 1`
`rArr ((n+1)/(n)) x gt 1 and ((n)/(n+1)) x lt 1`
`rArr x gt (n)/(n+1) and x lt (n+1)/(n)`
`rArr (n)/(n+1) lt x lt (n +1)/(n) rArr x in ((n)/(n+1), (n+1)/(1))`.