Question 1150903
There are two components here, how many receive the new email in a round, and the total of students that have been informed.

Each time an email is sent, it reaches {{{twice}}} as many people as the time before.  So the pattern is {{{2}}},{{{ 4}}},{{{ 8}}}, {{{16}}}, {{{32}}}, {{{64}}}, {{{128}}},{{{ 256}}}, {{{512}}}, etc.  

This is {{{2^n}}}, where {{{n}}} is a particular {{{round}}} of mailing.

Now when this starts, Vanessa knows, that's {{{1}}} person.  After the first mailing, it's {{{2+1}}} people that know, next it's {{{4+2+1}}}, then{{{ 8+4+2+1}}}, etc.  You might recognize this as {{{2^(n+1) - 1}}}.


a)  How many people receive it during the 7th mailing?

use {{{2^n}}}, where {{{n}}} is a particular {{{round}}} of mailing

so, in your case {{{n=7}}}

{{{2^n=2^7 = 128 }}}people


b) Which mailing will be sent to {{{500}}} people?

Since we already know that {{{2^7 = 128}}}, let's keep going from there.

{{{2^7=128}}}

{{{2^8 = 128 * 2 = 256}}}

{{{2^9= 256 *2 = 512}}}

Since we have exceeded {{{500}}}, we stop there. 

So, we know that since the power is {{{9}}}, the email has been sent {{{9}}} times.

c) There are {{{500 }}}students in the school, so how many does it take to send the mail to all of them?

This question was already answered in part b). The answer is {{{512}}}