Answer:
After n passing, the total number of people knowing the secret [tex]= 2^n[/tex].
After 10 passing, 1024 peoples know the secret.
Step-by-step explanation:
Given that each friend can pass the secret to 2 people.
On 1st pass:
Total number of people knowing the secret = 1+1=2
On 2nd pass:
Total number of people knowing the secret = 1+1+1x2=4
On 3rd pass:
Total number of people knowing the secret = 1+1+2+2x2=8
This can be written as [tex]1+1+2+2^2=1+1+2+2^{3-1}=8[/tex]
Or, [tex]1+S_3=8[/tex], where [tex]S_3=1+2+2^{3-1}[/tex].
So, for the [tex]n^{th}[/tex] pass:
Total number of people knowing the secret = [tex]1+1+2+2^2+\cdots+2^{n-1}=1+S_n, where, S_n = 1+2+2^2+\cdots+2^{n-1}[/tex].
As [tex]S_n[/tex] is the sum of geometric progression of n terms having the first term, [tex]a_1=1[/tex], and the common radio [tex]r=2[/tex].
So, [tex]S_n = \frac{a_1(r^n-1)}{r-1}[/tex]
[tex]\Rightarrow S_n =\frac{1(2^n-1)}{2-1}=2^n-1[/tex].
Hence, after n passing, the total number of people knowing the secret
[tex]=1+S_n = 1+2^n-1=2^n.[/tex]
After, 10 passing, put n=10.
Total number pf peoples know the secret [tex]= 2^{10}=1024.[/tex]
