If for positive integers r1,n2, the coefficients of the (3r)th and (r+2)th powers of x i…

Question

If for positive integers $r>1,n>2$, the coefficients of the $(3r)$th and $(r+2)$th powers of $x$ in the expansion of $(1+x)^{2n}$ are equal, then $n$ is equal to:

Accepted Answer

Given ${}^{2n}C_{r+2}={}^{2n}C_{3r}\Rightarrow(3r)!(2n-3r)!=(r+2)!(2n-r-2)!$

Testing $n=2r+1$: LHS $=(3r)!(4r+2-3r)!=(3r)!(r+2)!$; RHS $=(r+2)!(3r)!$. These match, confirming $n=2r+1$.

Solutions · 1