If three consecutive coefficients in the expansion of \((x+y)^n\) are in the ratio \(1:9:63\), then the value of \(n\) is

The value of $n$ is 39. If the three coefficients are indexed by $r,r+1,r+2$, their successive ratios give $n-r=9(r+1)$ and $n-r-1=7(r+2)$. Equating the two expressions for $n-r$ gives $r=3$, then $n=r+36=39$. Stopping at $n-r$ gives 36.