Question

1/2*5+1/5*8+1/8*11+...+1/77*80

Answer 1

[tex]\dfrac{1}{2\cdot 5}+\dfrac{1}{5\cdot 8}+\ldots+ \dfrac{1}{77\cdot 80}=\\ \dfrac{1}{3}\cdot \left(\dfrac{3}{2\cdot 5}+\dfrac{3}{5\cdot 8}+\ldots +\dfrac{3}{77\cdot 80}\right)=\\ \dfrac{1}{3}\cdot \left(\dfrac{5-2}{2\cdot 5}+\dfrac{8-5}{8\cdot 5}+\ldots +\dfrac{80-77}{77\cdot 80}\right)=\\ \dfrac{1}{3}\cdot \left(\dfrac{1}{2}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{8}+\ldots +\dfrac{1}{77}-\dfrac{1}{80}\right)=\\ \dfrac{1}{3}\cdot \left(\dfrac{1}{2}-\dfrac{1}{80}\right)=\\ [/tex]
$\dfrac{1}{3}\cdot \dfrac{39}{80}=\boxed{\dfrac{13}{80}}$