What type of sequence do the distance between successive bounces if a ball represent.
You have a situation here where each bounce is 80% of the previous bounce.
i.e. each bounce is a specific multiple of the previous bounce.
This, in effect, is the very definition of a gemetric series.
Answer: the type of sequence is a geometric sequence
A geometric series would be, for example,
b_(n+1) = b_n*bf, n = 1,2,3,... where bf is the bounce factor, or bf = 80% = 0.8
This could also be written as,
b_(n+1) = b_1 * (bf)^n, n = 1,2,3,...
If you wanted to find the total distance travelled after n bounces, this would be,
S_n = b_1 * (1 - bf^n)/(1 - bf), n = 1,2,3,...
And after an infinite number of bounces, the total distance travelled would be,
S_infinity = b_1 * 1/(1 - bf) = b1 * 1/(1 - 0.8) = b1 * 1/0.2 = 5*b_1
S_infinity = 5*b_1 = 10m