Question

After this first encounter, the bird then turns
around and flies from the runner back to the
finish line, turns around again and flies back
to the runner. The bird repeats the back and
forth trips until the runner reaches the finish
line.
How far does the bird travel from the beginning (including the distance traveled to the
first encounter)?

Answer 1

Repeating this scenario a second time the distance for the runner to travel after the second encounter is

$$L_2=\frac{3}{5}L_1=\left(\frac{3}{5}\right)^2L$$

and the third time is

$$L_3=\frac{3}{5}L_2=\left(\frac{3}{5}\right)^3L$$

and the i-th time is

$$L_i=\frac{3}{5}L_{i-1}=\left(\frac{3}{5}\right)^iL$$

 The distance the bird travels between the (i 1)-th and i-th time is

$$d_i=\frac{8}{5}L\left(\frac{3}{5}\right)^i$$

 and summing over all terms

$$d=\sum_{i=0}^\infty d_i=\frac{8}{5}L\sum_{i=0}^\infty \left(\frac{3}{5}\right)^i=\frac{8}{5}L\frac{5}{2}\\=4L=4(2.7)=10.8\ km$$