# Calculate the sum

### Calculate the sum

Calculate the sum:
$\frac{1}{1\cdot2}+\frac{1}{2\cdot2}+\frac{1}{3\cdot2}+...+\frac{1}{n(n+1)}$

Math Tutor

Posts: 316
Joined: Sun Oct 09, 2005 11:37 am

### Re: Calculate the sum

Are you sure you typed the question correctly? I don't see the pattern of your terms.

The start of your sum are the first three values of the sequence $\frac{1}{n\cdot 2}$ which doesn't match the last term $\frac{1}{n(n+1)}$.

If you meant $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \ldots + \frac{1}{n(n+1)}$
Spoiler: show
$\frac{1}{r(r+1)} = \frac{1}{r} - \frac{1}{r+1}$
so
$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \ldots + \frac{1}{n(n+1)} = (\frac{1}{1} - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + (\frac{1}{3} - \frac{1}{4}) + \ldots + (\frac{1}{n} - \frac{1}{n+1}) = 1 - \frac{1}{n+1}$

R. Baber
Guest

Perfect!
Math Tutor