Total height = 10 m
Drops 10 m
1st bounce = 0.8*10 = 8 m up
+ 8 m down
2nd bounce = 0.8*8 = 6.4 m up
+ 6.4 m down
3rd bounce = 0.8*6.4 = 5.12 m up
+ 5.12 m down
....
9th bounce = 10(0.8)^9 = 1.34 m up
+ 1.34 m down
See the pattern? each bounce multiplies another 0.8 ; count the distance twice for returning to the ground
10, 8, 6.4, 5.12, ... 1.34
term in sequence = 10(0.8)^(n-1)
where n = {1,2,3,...10}
Total meters traveled
= ∑ 10(0.8)^0 + 2[10(0.8)^1 + 10(0.8)^2 + ..... 10(0.8)^9]
multiply sum by 2 then subtract the extra 10 m
Sum of a geometric sequence for the bounces
where n in (n-1) [tex]S = a( \frac{1- r^{n} }{1-r} ) \\ \\ S = 10 ( \frac{1- 0.8^{10} }{1-0.8} ) \\ \\ S = 44.63
\\ \\ 2*44.63 = 89.26 \\ \\ 89.26 - 10 = 79.3 m [/tex]would be 10 for the number of bounces before 10th bounce, "a" is 10.
