# moving fraction into denominator

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

Image of equation

• I can't figure out what the expression in the image is supposed to be. Please type it. Here's a MathJax tutorial – saulspatz May 16 at 11:37

The second and third steps in the transformation: $$\frac{k(k+1)}2+k+1=\frac{k(k+1)}2+\frac{2(k+1)}2=\frac{k(k+1)+2(k+1)}2= \frac{(k+2)(k+1)}2$$ are called "reducing to the common denominator", which essentially says: