# (School) How long does it take to send an e-mail to 25000 people

My math teacher gave us a task:

There are 30 students each student sends an e-mail with a message to 2 other people. And everybody who recieved this e-mail sends it again to 2 other people and so on.

The Question: How much rounds does it take to inform 25000 people.

For me its obvious that it takes 9 rounds. But my Math teacher doesn't share this opinion he says that it takes 10 rounds. And to proof his opinion he uses the formula: f(x)=30*2^x . but I think this formula is basically wrong and this is the right formula: f(x)=((30*2^x)*2)-30 .

Chart to prove my opinion:

So my question is who is right?

• Are all those that received it sending it every round, so that the ones that got it the first time send it many times? In that case it seems that it should be $30 \cdot 3^x$ instead: each participating "person" becomes 3 people (themselves plus two more) who then participate in all subsequent rounds. – Ian Feb 4 '18 at 19:48
• No only the people who got the e-mail last round will send it to two other people. So in the first round there are 30 students that know the message + the people who got the message by the students. 30 + 60 = 90 . And in the second round there are the students + the people who got the message by the students + the people who got the message by the people who got the message by the students. 30 + 60 + 120 = 210 – Epig Feb 4 '18 at 19:51
• Ah, then after $n$ rounds you should have $30 \cdot \sum_{i=0}^n 2^i = 30 \frac{1-2^{n+1}}{1-2}=30 \cdot(2^{n+1}-1)$ which is your formula. – Ian Feb 4 '18 at 19:55
• This is a very nice question, +1 from me. I suspect that you and your professor are interpreting the problem differently, that you're both giving correct answers, albeit to slightly different problems. – pjs36 Feb 4 '18 at 20:00
• Thank you very much for all your comments, I think it will help a lot. – Epig Feb 4 '18 at 21:33