population manhattan word problem

[vineeshasatwani1]

Question from the 3rd manhattan guide -word problem
The population of grasshoppers doubles in a particular field every year. Approximately how many years will it take the population to grow from 2,000 grasshoppers to 1,000,000 or more?

i am trying to solve it using geometric progression where b=a *(r^(n-1)) :where b is the nth term and a is the first
i am getting 1000000=2000(2^n-1)
which on solving gives me 2^(n-1)=500
i want to know how should i convert 500 to the power of 2 inorder to easily solve for n.what i did basically is i know 2^9 is 512 which is approximately equal to 500 so i considered my new eqn to be
2^n-1 = 2^9 which gives me n value to be 10.
but the real answer is 9

please help me on where i am going wrong and also a better way to convert a number to the nearest exponent like here i converted 500 in power of 2 as 2^9 ,is there any method to do the same.

[RonPurewal]

Your problem here is that the population is 2000 at time zero, not when the time is 1 year. So, all of your time values are going to have an extra year.

Two things you could do:

* You could just subtract the extra year back off at the end of the problem.

* You could write the original formula with the value 2000 corresponding to n = 0, rather than to n = 1.