ANSWERS: 2
  • ill sleep on it
  • You can't get an exact closed form solution, but you can get an approximation. divide by 100: 4 * 1.1^(n-1) > n + 3 Roughly sketch both sides. The right side is a straight line, the left is an exponential. When n=1 the two sides are equal, but when n=2 the inequality is false: 4.4 < 5 so the left side is below the right on a graph, but because of the shape of the exponential the inequality is true for n<1 and at some point with n>2 it becomes true again. How to find it? Since there is no exact solution, need to use methods from numerical analysis. One such method is Newton-Raphson for solving when something = 0. You pick a point near the answer, find the gradient at the current point and pretend the function is a straight line, follow the straight line until it hits zero. Then repeat. So an equation which is zero is: 4 * 1.1^(n-1) - n - 3 = 0 The gradient of the left hand side is: 4 * 1.1^(n-1) * ln(1.1) - 1 Pick a value for n. I pick 50 because the function behaves similarly everywhere to the right of the crossing point and I expect it to be less than this. Value of function at n = 50 is 373.4 Rate of increase is 39.7 So I might be about 373.4/39.7 too far forwards. So try n=40 Value of function at n = 40 is 121.6 Rate of increase is 14.7 So try n=30 Value of function at n = 30 is 30.5 Rate of increase is 5.05 which suggests I'm about 6 too far forwards but based on how I didn't go far enough last time, I'll try n=20 Value of function at n = 20 is 1.46 Rate of increase is 1.33 Now I'm close I need to trust Newton, because in most cases, near to the solution, it doubles the precision each time you do it. 1.46/1.33 = about 1.10 Value of function at n=18.9 is 0.129 Rate of increase is 1.099 0.129/1.099 = about 0.117 Value of function at n=18.783 is 0.001423 Rate of increase is 1.076 0.001423/1.076 = about 0.001322 Value of function at n=18.781678 is 0.000002849 Repeat for more accuracy but already it looks like it's 18.78 something.

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