Question 1150903
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Part a)
During the nth mailing, 2^n students will be messaged. So for the first mailing (n=1) we have 2^n = 2^1 = 2 students messaged (the two vice presidents). Then for round two (n=2), we have 2^n = 2^2 = 4 students messaged in this round.


For the 7th mailing, plug n = 7 into 2^n to get 2^n = 2^7 = 128. 


Answer: <font color=red size=4>128</font>

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Part b)


Think backwards from part a). We want to aim for 500 students in a single round of messaging. The question is "what value of n will make 2^n equal 500?". 


We want to solve
2^n = 500


Use logarithms to answer that question. You will need your calculator.
2^n = 500
Log(2^n) = Log(500)
n*Log(2) = Log(500)
n = Log(500)/Log(2)
n = 8.96578428466209


Note how if n = 8, then
2^n = 2^8 = 256
and when n = 9, we have
2^n = 2^9 = 512
so as expected, the solution is somewhere in between n = 8 and n = 9


Using n = 8 is too small, so we have to go with n = 9


Answer: <font color=red size=4>9th round</font>

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Part c)


Here is a list of powers of 2
1,2,4,8,16,32,64,128,256,512,...


Let's compute partial sums
1+2 = 3
1+2+4 = 7
1+2+4+8 = 15
1+2+4+8+16 = 31
1+2+4+8+16+32 = 63
1+2+4+8+16+32+64 = 127
1+2+4+8+16+32+64+128 = 255
1+2+4+8+16+32+64+128+256 = 511
Each sum is 1 less than the next power of 2. The general formula is
*[Tex \LARGE \displaystyle \sum_{k=0}^{n}2^k = 2^{n+1}-1]


What this means is that we can quickly compute how many students have been messaged in total (rather than just on an individual round).


Set 2^(n+1) - 1 equal to 500 and solve for n.


2^(n+1) - 1 = 500
2^(n+1) = 500+1
2^(n+1) = 501
Log(2^(n+1)) = Log(501)
(n+1)*Log(2) = Log(501)
n+1 = Log(501)/Log(2)
n+1 = 8.9686667931952
n = 8.9686667931952-1
n = 7.9686667931952
That rounds to n = 8


By round 8, a total of 
2^(n+1)-1 = 2^(8+1)-1 = 511
people have been messaged
If we ignore Vanessa, then 511-1 = 510 people have been messaged


Of course this school has 500 students and not 510 or 511. The same value of n is the answer however because in the 8th round 2^8 = 256 people are messaged. Subtracting 11 from this count is still a positive value.


Answer: <font color=red size=4>8th round</font>


As you can see, there is no need to have a 9th round in which 512 people are messaged because the entire school will be notified by the 8th round
1+2+4+8+16+32+64+128+256 = 511
note that 2^n = 2^8 = 256

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