If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1+x)n+5 are in the ratio 5:10:14, then the largest coefficient in this expansion is:
Solutions · 1
Official solution
Let the three consecutive coefficients be n+5Cr,n+5Cr+1,n+5Cr+2.
\[\frac{{}^{n+5}C_r}{{}^{n+5}C_{r+1}}=\frac{5}{10}=\frac12\quad\Rightarrow\quad\frac{r+1}{n+5-r}=\frac12\quad\Rightarrow\quad2r+2=n+5-r\quad\Rightarrow\quad3r=n+3\quad\text{...(i)}\]
\[\frac{{}^{n+5}C_{r+1}}{{}^{n+5}C_{r+2}}=\frac{10}{14}=\frac57\quad\Rightarrow\quad\frac{r+2}{n+4-r}=\frac57\quad\Rightarrow\quad7r+14=5n+20-5r\quad\Rightarrow\quad12r=5n+6\quad\text{...(ii)}\]
Solving (i),(ii): r=4,n=6, so n+5=11.
Largest coefficient: middle term of (1+x)11 is (511)=(611)=462.