Discussion Forum

The coefficient of three consecutive terms  of  (1+x)ⁿ⁺⁵ are in the ratio 5:10:14 , then , n is equal to

Answer:

Explanation:

Let the  three consecutive terms in (1+x) ⁿ⁺⁵   be  t_r , t_r+1, t_r+2 . Having  coefficients   

$^{n+5}C_{r-1},^{n+5}C_{r}, ^{n+5}C_{r+1}$   given,  $^{n+5}C_{r-1}:^{n+5}C_{r}: ^{n+5}C_{r+1}$ =5:10:14

$\frac{^{n+5}C_{r}}{^{n+5}C_{r-1}}=\frac{10}{5}$

  and   $\frac{^{n+5}C_{r+1}}{^{n+5}C_{r}}=\frac{14}{10}$

 $\Rightarrow$    $\frac{n+5-(r-1)}{r}=2$

  and    $\frac{n-r+5}{r+1}=\frac{7}{5}$

$\Rightarrow$    n-r+6=2r

 and    5n-5r+25=7r+7

 $\Rightarrow$     n+6=3r

 and 5n+18=12 r

  $\therefore$    $\frac{n+6}{3}=\frac{5n+18}{12}$

$\Rightarrow$   4n+24=5n+18

$\Rightarrow$   n=6