Question

If the coefficient of three consecutive terms in the expansion of $(1+x{)}^n$ be 165 , 330 , and 462. Find n

Answer 1

Let the successive terms be $T_{r-1},T_r,T_{r+1}$ whose coefficient are 165 , 330 , and 462 respectively

$\frac{C_r}{C_{r-1}}=\frac{n-r+1}{r}$
$\therefore \frac{n-r+1}{r}=\frac{330}{165}=\frac{2}{1}$
$\therefore n-3r+1=0$

$\frac{C_{r+1}}{C_r}=\frac{n-r}{r+1}$

$\therefore \frac{n-r}{r+1}=\frac{462}{330}=\frac{7}{5}$

$\therefore 5n-12r-7=0$

$n=11$
$\therefore r=4$