Step-by-step explanation:
a 75% rebound ratio means that the rebounding back reaches 75% of the height of the previous drop.
75% = 3/4
a geometric series means that every term is created by multiplying the previous term by a certain factor. this factor is this mentioned 3/4.
each term (except s0, the start value) stands for the hight of the bounce.
s0 = k
s1 = s0 × 3/4 = k × 3/4
s2 = s1 × 3/4 = k × 3/4 × 3/4 = k × (3/4)²
sn = sn-1 × 3/4 = k × (3/4)^n
for k = 235ft
the 6th bounce is
s6 = 235 × (3/4)⁶ = 41.82495117... ft
the total distance by the ball is
2× the sum of the terms until s11 (the last bounce before the ball strikes the ground a 12th time) - s0.
why ? for s1 to s11 the ball has to go up and down, so the height of the bounce is traveled twice. but s0 happens only once (the first drop). so, when we multiply the whole sum by 2, we need to deduct s0 one time again.
the sum of all bounces s0 to s11 (n = 11) is
s = s0 × (1 - r^(n+1))/(1 - r)
where r is the multiplication factor (here 3/4).
s = 235 × (1 - (3/4)¹²)/(1 - 3/4) =
= 235 × (1 - (3/4)¹²)/(1/4) = 235 × (1 -(3/4)¹²) × 4 =
= 910.2242291... ft
2×s - s0 = 1,585.448458... ft
