Question

If a, b, c, d be four consecutive coefficient in the binomial expansion of $(1+x{)}^n$ , then value of the expression
$[{\left[\frac{b}{b+c}\right]}^2-\frac{ac}{(a+b)(c+d)}](wherex>0andn\in N)$ is

• A. positive
• B. negative
• C. zero
• D. depend on n

Answer 1

The correct option is A positive

$a{=}^n{C}_{r-1},b{=}^n{C}_{r}c{=}^n{C}_{r+1}d{=}^n{C}_{r+2}$

$a+b{=}^{n+1}{C}_{r}b+c{=}^{n+1}{C}_{r+1}c+d{=}^{n+1}{C}_{r+2}$

$\frac{a+b}{a}=\frac{n+1}{r}$

$\Rightarrow \frac{a}{a+b}=\frac{r}{n+1},\frac{b}{b+c}=\frac{r+1}{n+1},\frac{c}{c+d}=\frac{r+2}{n+1}$

$\therefore \frac{a}{a+b},\frac{b}{b+c},\frac{c}{c+d}$ are in A.P.

AM> GM

$\frac{b}{b+c}>\sqrt{\frac{ac}{(a+b)(c+d)}}\Rightarrow {\left[\frac{b}{b+c}\right]}^2-\frac{ac}{(a+b)(c+d)}>0$