If $a_{n+1} - a_n = \frac{1}{n+1}$ how to find $a_n$?

Question

Suppose you have $a_0=0$ and $\forall n > 0$ $$a_{n+1} - a_n = \frac{1}{n+1}$$, how do I find $a_n$?

I checked the first values, and this is what I got:

$$a_0=0$$ $$a_1=1 $$ $$a_2 = \frac{3}{2} $$ and $$a_3= \frac{11}{6} $$

I don't see any obvious relation. Would there be a method to solve such relations?

Answer 1

$$a_n= \sum_{k=1}^{n}(a_k-a_{k-1})=1+1/2+....+1/n$$

Answer 2

Just write

$$a_{n+1}=a_n+\frac1{n+1}$$

Then $a_1=a_0+1$, $a_2=a_1+\frac12=a_0+1+\frac12$ and by induction we have that

$$a_n=a_0+\sum_{k=1}^n\frac1k$$