Question 957846
We can prove this equation using mathematical induction.
Let n=1
LHS=1*1!=1
RHS=2!-1=2-1=1
LHS=RHS
The equation holds true for n=1
We will assume that the equation holds true for n=k
1*1!+2*2!+.........+k*k!=(k+1)!-1......equation 1
For n=k+1 we have
LHS=1*1!+2*2!+.........+k*k!+(k+1)*(k+1)!
   =(k+1)!-1+(k+1)*(k+1)!
   =(k+1)!(k+1+1)-1
   =(k+2)(k+1)!-1
   =RHS
Therefore the equation holds true for all N.