Question

If a, b, c, d are any four consecutive coefficients in the expansion of $(1+x{)}^n$, then $\frac{a}{a+b},\frac{b}{b+c},\frac{c}{c+d}$ are in.

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P

Answer 1

Answer: (A)

A.P.

Let $a{=}^n{C}_r$

$b{=}^n{C}_{r+1}$

$c{=}^n{C}_{r+2}$

$d{=}^n{C}_{r+3}$

$\therefore \frac{a}{a+b}=\frac{{}^n{C}_r}{{}^n{C}_r{+}^n{C}_{r+1}}=\frac{r+1}{n+1}$

$\therefore \frac{b}{b+c}=\frac{{}^n{C}_{r+1}}{{}^n{C}_{r+1}{+}^n{C}_{r+2}}=\frac{r+2}{n+1}$

$\therefore \frac{c}{c+d}=\frac{{}^n{C}_{r+2}}{{}^n{C}_{r+2}{+}^n{C}_{r+3}}=\frac{r+3}{n+1}$

$\therefore \frac{a}{a+b},\frac{b}{b+c},\frac{c}{c+d}$ are in A.P. with common difference $d=\frac{1}{n+1}$.