Evaluate 1/(1+2) + 1/(1+2+3) + 1/(1+2+3+4)... + 1/(1+2+3+4...+100)
This is a math competition question that I have tried solving for a long time as I can't use a calculator. What is the simplest way of solving this question without using a calculator?
Hint:
$$\dfrac1{\sum_{r=1}^nr}=\dfrac2{n(n+1)}=\dfrac{2(n+1-n)}{n(n+1)}=f(n)-f(n+1)$$ where $f(r)=\dfrac2r$
Use Telescoping series