Compute: $\frac3{7\cdot2}+\frac3{7\cdot12}+\dots$

Question

Compute: $$\frac3{7\cdot2}+\frac3{7\cdot12}+\frac3{17\cdot12}+\dots+\frac3{2017\cdot2012}$$ I couldn't really find the pattern in this one. I tried evaluating the first two terms which was $\frac14$, but I don't know how to progress from there.

Answer 1

Swapping the factors in alternate fractions, then collecting the 3's, gives $$3\left(\frac1{2\cdot7}+\frac1{7\cdot12}+\dots+\frac1{2012\cdot2017}\right)$$ This is therefore a standard telescoping sum, as $$\frac1{2\cdot7}=\frac15\left(\frac12-\frac17\right)$$ $$\frac1{7\cdot12}=\frac15\left(\frac17-\frac1{12}\right)\dots$$ Therefore the sum collapses to $$\frac35\left(\frac12-\frac1{2017}\right)=\frac{1209}{4034}$$