If `n` is an even positive integer, then find the value of `x` if the greatest term in the expansion of `1+x` ^n may have the greatest coefficient also.

Answer

`x gt (n)/(n+2)` and `x lt (n+2)/(n)`

Explanation

If n is even, the greatest coefficient is `.^(n)c_(n//2)`. Therefore, the greatest term is `.^(n)C_(n//2)x^(n//2)`.
`:. .^(n)C_(n//2)x^(n//2)gt.^(n)C_((n//2)-1)x^((n-2)//2)` and `.^(n)C_(n//2)x^(n//2) gt .^(n)C_((n//2)+1)x^((n//2)+1)`
`rArr (n-n/2+1)/((n)/(2)) xgt 1` and `((n)/(2))/((n)/(2) + 1) x lt 1`
`rArr xgt ((n)/(2))/(n/2+1)` and `x lt ((n)/(2)+1)/((n)/(2))`
`rArr x ge (n)/(n+2)` and `x lt (n+2)/(n)`