A question about math trees?

Question

There is this question. There is a secret, only one person knows it. He tells 2 people, and thats it. Then, the two people tell 2 other people. It keeps on going like this. How many times is the the secret going to get to 100 people. I know how many it is for 100. It is 7 times. The only way to do this is to do 1+2+4+8+16+32+64. You get to more than 100, but you got to it.Now, my question is, is there any way you can just make a formula, like100-93=7, to solve this problem easier. If there is know way, can you state the theory why this is impossible.

Answer 1

Yes
Here is the formula
2^n - 1 = 100
or 2^n = 100 + 1
or 2^n = 101
or n= log( to the base 2) 101
or n = (log 101 )/ log 2
or n = 6.65
As n cant be in fraction so rounding it we get n = 7
U can directly find the answer by using the formula
n = log(numer of people +1) / log 2

Answer 2

You're basically dealing with the function 2^n where n is a nonnegative integer.
Initially 1 = 2^0 person knowns the secret.
This person tells the secret to 2 = 2^1 people.
Each of the two persons tells the secret to another two people.
So 2^1 + 2^1 = 2^2 new people know the secret now.
The idea is that, at the nth stage, 2^(n-1) new people know the secret. This means that the total number of people who know the secret at the nth stage is: 2^0+2^1+...+2^(n-1)
Now, 2^0+2^1+...+2^(n-1) = (1-2^n)/(1-2) = 2^n - 1
To find out the number of stages the secret takes to get to a 100 people, you simply have to solve for n the following inequality:
2^n - 1 => 100
2^n => 101
n => log(101) where the logarithm is taken base 2.
At the very beginning, n was assumed to be an integer. So n would be the least integer greater than or equal to log(101).
log(101) ~ 6.65. Clearly, the least integer greater than or equal to 6.65 is 7.

Answer 3

Take the number of people.
Add one.
Use your calculator to find the LOG of this number.
Divide the result by the LOG of 2 (0.30103).
Round UP to the nearest integer.

Example: 100
Add one: 101
LOG: 2.00432
Divide by 0.30103: 6.65821
Round up: 7

Example: 255
Add one: 256
LOG: 2.40824
Divide by 0.30103: 8.00000
Round up: 8

Example: 50000
Add one: 50001
LOG: 4.69898
Divide by 0.30103: 15.60967
Round up: 16

Answer 4

Yes

2^(n)-1
Where n is the generation of people concerned. The original person is generation 1.