Geometric Sequences math homework

Question

When an object is allowed to fall freely near the surface of the earth, the gravitational pull is such that the object falls 16 ft in the first second, 48 ft in the next second, 80 ft in the next second, and so on.

(a) Find the total distance a ball falls in 6 s.

(b) Find a formula for the total distance a ball falls in n seconds.

Answer 1

The actual total distance fallen is 16, 16+48, 16+48+80, that is 16, 64, 144.

(a) If we write the series as multiples of 16 we have 1, 4, 9. These are the squares of the natural numbers so the total distance fallen is 16n^2 where n is the number of seconds. The difference between the nth second and the (n-1)th second is 16(n^2-(n^2-2n+1))=16(2n-1). Therefore the total distance fallen in 6 seconds is 16*36=576 ft, but the distance fallen between n=5 and 6 is 16(12-1)=176 ft.

(b) The total distance fallen in n seconds is 16n^2.