Geometric Sequences: What is $r^n$, and how do I find $r$?

Question

I've been tasked to find the summation $$\sum_{x=1}^{100} \frac{1}{x(x+1)}$$ without manually summing each term. I've looked at all the shortcuts I can find online, and I was only able to find shortcuts for constant multiples, squared terms, addition/subtraction, and the like.

I finally found something about geometric sequences, but I'm having trouble understanding it. How do I find $r^n$ and $r$?

I tried to write out the first few terms. I got $\frac{1}{2}$, $\frac{1}{6}$, $\frac{1}{12}$, and $\frac{1}{20}$. I can't really see a common ratio other than the formula itself. Maybe this isn't a geometric series? If not, how can I find this summation formulaically?

Answer 1

This is not a geometric series. One way of solving it is to write$$\sum_{x=1}^{100} \frac{1}{x(x+1)} = \sum_{x=1}^{100} \frac{1}{x} - \frac{1}{x+1} = 1 - \frac{1}{101} = \frac{100}{101}$$

Answer 2

$$\sum _{x=1}^{100} \frac{1}{x(x+1)}=\\\sum _{x=1}^{100} \left(\frac{1}{x}-\frac{1}{x+1}\right)=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\ldots+\frac{1}{98}-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}+\frac1{100}-\frac1{101}=\\ 1-\frac{1}{101}=\frac{100}{101}$$