Hello Mathematics Stackexchange I had a quick question. I do sincerely apologize if this type of question was asked before. Im having trouble simplifying this fraction specifically I am not sure how that second term was multiplied by 2 and the fraction (specifically the 2 was moved into the denominator).
P.S - Sorry Im not the most technologically advanced, I tried putting it in the desired format that was common on this site but ran into issues, regardless I wrote out the steps here.
Original equation : S(k)= 1/2k(k+1)
The format we are trying to get to : S(k+1)= 1/2(k+1)((k+1)+1)
= 1+2+...+k+(k+1) = S(k)+(k+1) = 1/2k(k+1)+(k+1) = (k(k+1)+2(k+1))/2 = ((k+1)(k+2))/2 = 1/2(k+1)((k+1)+1)
Now I get the induction steps but I've seen to forgot the basic rule that allows the 1/2 to moved into the denominator so I was just wondering why this was allowed(for future reference).
I have attached the Image below