Question

Your brother tells you a secret. You see no harm in telling two friends. After this second

"passing" of the secret, 4 people now know the secret (your brother, you and two friends). If
each of these friends now tells two new people, after the third "passing of the secret, eight
people will know. Write an equation to express the "passing" of the secret. If this pattern of
spreading the secret continues, how many people will know the secret after 10 such
"passings"?

Answer 1

After n passing, the total number of people knowing the secret
`= 2^n`.

After 10 passing, 1024 peoples know the secret.

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
`1+1+2+2^2=1+1+2+2^(3-1)=8`

Or,
`1+S_3=8`, where
`S_3=1+2+2^(3-1)`.

So, for the
`n^(th)` pass:

Total number of people knowing the secret =
`1+1+2+2^2+\cdots+2^(n-1)=1+S_n, where, S_n = 1+2+2^2+\cdots+2^(n-1)`.

As
`S_n` is the sum of geometric progression of n terms having the first term,
`a_1=1`, and the common radio
`r=2`.

So,
`S_n = (a_1(r^n-1))/(r-1)`

`\Rightarrow S_n =(1(2^n-1))/(2-1)=2^n-1`.

Hence, after n passing, the total number of people knowing the secret

`=1+S_n = 1+2^n-1=2^n.`

After, 10 passing, put n=10.

Total number pf peoples know the secret
`= 2^(10)=1024.`