simplifying equations

Question

I have been trying to study analysis of algorithms with mathematical induction yet I found my algebra skills are very poor. So now I began restuddiing algebra (factoring, destributive property, simplifying, expanding) so I can handle mathematical equations better. But no matter what I do, I can't seem to explain how the following equation work.

$$ \frac{n(n+1)}{2}+(n+1) = \frac{(n+1)(n+2)}{2} $$

I think $$ \frac{n(n+1)}{2}+(n+1) $$ have been simplified to

$$ \frac{(n+1)(n+2)}{2} $$

I have a feeling I am thinking something stupid, but I really don't understand how the right side of the equation have been simplified.

• Can anyone explain how the formula got simplified?
• What part of algebra should I study to understand equations like this?

Answer 1

Note that both expressions have the common factor $(n+1)$, which we can factor out. Hence we write $$\frac{n(n+1)}{2}+(n+1)=(n+1)\left[\frac{n}{2}+1\right]=(n+1)\left[\frac{n}{2}+\frac{2}{2}\right]=(n+1)\frac{n+2}{2}=\\=\frac{(n+1)(n+2)}{2}$$

Answer 2

user222031 got the easy one. If one knows that $$ \frac{n(n+1)}{2} = \sum_{k=1}^n k $$ (the sum of the first n integers is n * (n+1) / 2). Then it is also obvious, because another term ($n+1$) has been added to the sum, now summing the first n+1 integers, which means that n needs to be changed to n+1 in the result formula, giving (n+1)*(n+2)/2.