π Example. (Bungee Jumping). A bungee jumper falls 35 meters before his cord causes him to spring back up. He rebounds β of the distance after each fall.
a. Find the first five terms of the infinite sequence representing the vertical distance traveled by the bungee jumper. Include each drop and rebound distance as separate terms.
b. What is the total vertical distance the jumper travels before coming to rest? (Hint: Rewrite the infinite sequence suggested by in part (a.) as two infinite geometric sequences.)
β Solution:
a. The bungee jumper will fall 35 meters, spring back up β (35) or 14 meters, fall 14 meters, spring back up β (14) or 5.6 meters, fall 5.6 meters, spring back up β (5.6) or 2.24, and so on.
Therefore, the first five terms of the infinite sequence that represents the vertical distance traveled by the bungee jumper are 35, 14, 14, 5.6, and 5.6.
b. The series that corresponds to the infinite sequence 35, 14, 14, 5.6, 5.6, can be written as the sum of the two infinite geometric series: one series that represents the distance traveled when falling and one series that represents the distance traveled when springing back up.
Series 1 35+14+5.6+β―
Series 2 14+5.6+2.24+β―
Find the sum of each series.
Sum of an infinite geometric series formula
Therefore, the total vertical distance that the jumper travels is equal to SSeries 1+SSeries 2.
