Problema Solution
Mary's parents decided to give her an allowance every Saturday. The first Saturday they gave her $.01. For each successive week, they double the amount given the previous week. If Mary saves her allowance, in how many weeks will she be a millionaire?
Answer provided by our tutors
beginning : $0.01
after 1 week: 2*0.01
after 2 weeks: 2*2*0.01 = (2^2)*0.01
after 3 weeks: 2*2*2*0.01 = (2^3)*0.01
after n weeks: (2^n)*0.01
Mary will become a millionaire when the sum of all the money will be at least 1,000,000
We need to find the sum Sn of the terms of a geometric progression
a0 = 0.01, an = (2^n)*0.01 where the common ratio r = 2 and the scale factor a = 0.01
Sn = (a*(1 - r^(n +1))/(1 - r)
Sn = (0.01*(1 - 2^(n+1))/(1 - 2))
Sn = 0.01*(2^(n+1) - 1)
lets find for what n is Sn = 1,000,000 that is
0.01*(2^(n+1) - 1) = 10^6
2^(n+1) = 10^6/0.01
2^(n+1) = 10^8
(n+1) l0g 2 = 8
n = (8/l0g 2)
n = 26.67 weeks
since we need n integers such that Sn > 10^6 we conclude that in 27 weeks Mary will be a millionaire.