# precal

A pile driver pounds a steel column into the ground. On the first drive, the column is pounded 1.65 metres into the ground, and on each successive drive it moves 80% as far as it did on the previous drive. The least number of drives required to drive the column a total of 8.2 metres into the ground is _____.

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1. so you want
1.65 + 1.65(.8) + 1.65(.8)^2 + .. = 8.2

you could just keep adding until you reach 8.2 but that would be less elegant than using a geometric series

a = 1.65
r = .8
Sum(n) = 8.2
a(1 - r^n)/(1-r) = sum(n)
1.65(1 - .8^n)/(1-.8) = 8.2
(1-.8^n)/.2 = 8.2/1.65
1-.8^n = 8.2/1.65*.2
.8^n = 1 - 8.2/1.65*.2
n log .8 = log [1 - 8.2/1.65*.2]
n = log [1- 8.2/1.65*.2] / log .8 = 22.8

22 drives will not do it, so they will need 23 pile drivings

check:
Sum(22) = 1.65 (1-.8^22)/.2 = 8.189
sum(23) = 1.65(1-.8^23)/.2 = 8.2013

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