Show that $$\frac1{1\cdot2}+\frac1{2\cdot3}+\frac1{3\cdot4}+\cdots+\frac1{(n-1)\cdot n}=\frac{n-1}{n}.$$
I’m having a really hard time with this question - I can’t start it with one because you can’t divide by zero, and as I go further along I still am struggling. I started with two, so I did the base step, and the induction hypothesis step, how do you prove that K+1 is true? How do I do this process and what’s the answer?
Other approach
$$\frac{1}{1.2}=\color{green}{\frac{1}{1}}-\color{red}{\frac{1}{2}}$$
$$\frac{1}{2.3}=\color{red}{\frac{1}{2}}-\frac{1}{3}$$
$$\frac{1}{3.4}=\frac 13 -\frac 14$$ ...
$$\frac{1}{(n-1)n}=\frac{1}{n-1}-\color{green}{\frac{1}{n}}$$
if we sum, we find after telescoping $$1-\frac 1n=\frac{n-1}{n}$$