out.txt

Day	Due ($)	Total ($)
0	0.00	1000.00
1	100.00	1900.00
2	200.00	3600.00
3	400.00	6800.00
4	800.00	12800.00
5	1600.00	24000.00
6	3200.00	44800.00
7	6400.00	83200.00
8	12800.00	153600.00
9	25600.00	281600.00
10	51200.00	512000.00
11	102400.00	921600.00
12	204800.00	1638400.00
13	409600.00	2867200.00
14	819200.00	4915200.00
15	1638400.00	8192000.00
16	3276800.00	13107200.00
17	6553600.00	19660800.00
18	13107200.00	26214400.00
19	26214400.00	26214400.00
20	52428800.00	0.00
21	104857600.00	-104857600.00
22	209715200.00	-419430400.00
23	419430400.00	-1258291200.00
24	838860800.00	-3355443200.00
25	1677721600.00	-8388608000.00
26	3355443200.00	-20132659200.00
27	6710886400.00	-46976204800.00
28	13421772800.00	-107374182400.00
29	26843545600.00	-241591910400.00
30	53687091200.00	-536870912000.00

proj.js

var PERCENT = 0.1
  , INITIAL = 1000
  , GROWTH  = 2

// get Devil's due
function getDue (day) {
  if (day === 0) return 0
  return INITIAL * PERCENT * Math.pow(2, day - 1)
}

// Rather than using a recursive function, a loop is used to achieve O(n)
// Also saves stack space.
function getTotal (day) {
  var total = INITIAL
  // main loop for the integer part of the number
  for (var i = 1; i <= day; i ++) {
    total = total * GROWTH - getDue(i)
  }
  // fractional part of the number (part of the day)
  var frac = day % 1
  if (frac) {
    total += total * (GROWTH - 1) * frac - getDue(day)
  }
  return total
}

// binary search to find out what time exactly did it reach $1000.
var TARGET = INITIAL
  , ITE = 25
  , cur = 0
  , min = 19 // (19, 26214400)
  , max = 20 // (20,        0)
for (var i = 0; i < ITE; i ++) {
  var avg = (min + max) / 2
    , out = getTotal(avg)
  if (out > TARGET) min = avg
  else              max = avg
}