Question

( n+2)! + ( n+1)! = 15n!

Answer 1

$( n+2)! + ( n+1)! = 15n! \\ \\ ( n+2)(n+1) (n) ! + ( n+1)(n)! = 15n! \\ \\ n! \left[ ( n+2)(n+1) + ( n+1)}\right] = 15n! \\ \\ ( n+2)(n+1) + ( n+1)= 15n! :n!\\ \\ n^2+n+2n+2+n+1= 15\\ \\ n^2+n+2n+2+n+1-15= 0 \\ \\ n^2+4n-12= 0 \\ \\ (n-2)(n-6)=0 \\ \\ \boxed{n_1= 2\qquad\qquad n_2=6}$