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
Best Answer
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:
multiplication of the numerator and denominator by the same number does not change the value of the fraction.
sum of fractions with equal denominators is equal to the sum of the numerators divided by the (common) denominator.