Question

If `a,b,c,d` be four consecutive coefficients in the binomial expansion of `(1+x)^(n)`, then value of the expression `(((b)/(b+c))^(2)-(ac)/((a+b)(c+d)))` (where `x gt 0` and `n in N`) is
A. positive
B. negative
C. zero
D. depends on `n`

Answer 1

Correct Answer - A
`(a)` `a=^(n)C_(r-1)`, `b=^(n)C_(r )`, `c=^(n)C_(r+1)`, `d=^(n)C_(r+2)`
`a+b=^(n+1)C_(r )`
`b+c=^(n+1)C_(r+1)`
`c+d=^(n+1)C_(r+2)`
`(a+b)/(a)=(n+1)/(r )`
`implies(a)/(a+b)=(r )/(n+1)`, `(b)/(b+c)=(r+1)/(n+1)`, `(c )/(c+d)=(r+2)/(n+1)`
`:. (a)/(a+b)`, `(b)/(b+c)`, `(c )/(c+d)` are in `A.P.`
`A.M. gt G. M.`
`(b)/(b+c) gt sqrt((ac)/((a+b)(c+d)))`
`implies((b)/(b+c))^(2)-(ac)/((a+b)(c+d)) gt 0`