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