You receive a penny on January 1st, two cents the next day, four cents the next day, and so on, doubling every day. How long will it take to be a millionaire? Suppose you started wtih a nickel and doubled, how long before you were worth a million dollars?
2007-10-29 09:00:44 · 8 answers · asked by SM 2 in Science & Mathematics ➔ Mathematics
On day 1 you will receive 2^0 = 1 coin
On day 2 you will receive 2^1 = 2 coins (total = 3)
On day 3 you will receive 2^2 = 4 coins (total = 7)
On day 4 you will receive 2^3 = 8 coins (total = 15)
...
In general, on day n, you will receive 2^(n-1) coins for a total of 2^n - 1 coins.
You want to figure out the day when the accumulated amount (2^n - 1) is greater than or equal to 100 million coins (100 million pennies = 1 million dollars).
2^n - 1>= 100,000,000
2^n >= 100,000,001
Take the log base 2 of both sides.
n = log_2(100,000,001)
log(100,000,001) / log(2) = 26.5754248
n >= 26.5754248 days
27 days to be better than a millionaire with pennies.
The same logic for the nickel, but this time you only need 20 million nickels to make you a millionaire.
2^n - 1 >= 20,000,000
2^n > 20,000,001
n = log_2 (20,000,001)
log(20,000,001) / log(2) = 24.2534967
n >= 24.2534967
25 days to be better than a millionaire with nickels.
2007-10-29 09:41:15 · answer #1 · answered by Puzzling 7 · 0⤊ 0⤋
It's easy base 2 stuff. 2^(x-1)>50,000,000
For the nickel, you just modify it slightly to reflect that you're going by 5s, but still only doubling. 2^(x-1)>10,000,000.
That's it... you get 27 days starting with a penny and I believe 25 for the nickel.
I'm editing this to disagree with Louie's answer. You forget the accumulation. On day 1 you get 1 cent. On day 2, you get 2 more, so you have 3 cents. So in the first week you'll have
1, 3, 7, 15, 31, 63, and on the last day of that week, $1.27.
2007-10-29 16:25:09 · answer #2 · answered by Brent 2 · 0⤊ 1⤋
by January 28 (starting with a penny)
by January 26 (starting with a nickel)
use the geometric sequence formula
L = F * M^(N-1)
where
"F" = first term
"M" = the constant multiplier
"L" = the (value of the) last term "L"
"N" = the number of terms "N"
for the penny case, F=0.01, M = 2, L = 1000000
solve for N
N=27.57 = 28 days
for the nickel case, F=0.05, M = 2, L = 1000000
solving for N=25.25 = 26 days
to double check, you can just test it out
Day penny nickel
1 0.01 0.05
2 0.02 0.1
3 0.04 0.2
4 0.08 0.4
5 0.16 0.8
6 0.32 1.6
7 0.64 3.2
8 1.28 6.4
9 2.56 12.8
10 5.12 25.6
11 10.24 51.2
12 20.48 102.4
13 40.96 204.8
14 81.92 409.6
15 163.84 819.2
16 327.68 1638.4
17 655.36 3276.8
18 1310.72 6553.6
19 2621.44 13107.2
20 5242.88 26214.4
21 10485.76 52428.8
22 20971.52 104857.6
23 41943.04 209715.2
24 83886.08 419430.4
25 167772.16 838860.8
26 335544.32 1677721.6
27 671088.64
28 1342177.28
2007-10-29 16:18:42 · answer #3 · answered by bustedtaillights 4 · 1⤊ 2⤋
If you started with a penny, probably you would be a millionaire by the end of January, believe it or not. Try it.
2007-10-29 16:04:42 · answer #4 · answered by k-9fanatic 3 · 0⤊ 1⤋
Yikes, that's gonna take a while to do. Keep on doubling it two, four, six, eigth, ten, tweleve, etc...use a caculator so it will take less time.
2007-10-29 16:03:36 · answer #5 · answered by Dezzie. 3 · 0⤊ 3⤋
2379 years and 7 months. Dont get yer hopes up kid...oops, my bad. Thats if you get a penny every day. ^.^
2007-10-29 16:07:25 · answer #6 · answered by Stick-ninja 3 · 0⤊ 3⤋
i think you would do 1000000/2/2... im not sure tho.. and then for the nickel, you might do 1000000/5/2 but im not sure again. lol... im just kinda guessing.,
2007-10-29 16:04:24 · answer #7 · answered by Anonymous · 0⤊ 3⤋
Ha like i'd know!
2007-10-29 16:03:33 · answer #8 · answered by Anonymous · 1⤊ 2⤋