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

If a,b,c,d be four consecutive coefficients in the binomial expansion

1.	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
Answer» positive negative zero depends on `n` Solution :`(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`

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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

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