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1. The infinite sequence \(S=\{s_1,s_2,s_3,\ldots\}\) is defined by \(s_1=7\)and \(s_n=7^{s_{n-1}}\) for each integer \(n>1\). What is the remainder when \(s_{100}\) is divided by \(5\)?

 

2. What is the average of all positive integers that have three digits when written in base 5, but two digits when written in base 8? Write your answer in base 10.

 Dec 24, 2020
 #1
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Smallest two digit base 8 number  1 0      =  810

Largest two digit base 8 number    7 7     = 63 10

 

 

 Smallest 3 digit base 5    100 = 2510

Largest three digit base 5 = 444 =12410

 

So we need the average of base 10 numbers from 25   thru 63       1716/39 = 44 average10

 Dec 24, 2020
 #2
avatar+117725 
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2.

Possible base 5  integers   are   1(5)^2  + 0 + 0   = 25 base 10  to 4(5)^2 + 4(5)  + 4   = 124  base 10

 

Possible  base 8 integers  are   1(8)  + 0  =  8  base 10  to 7(8) + 7 =  63 base 10 

 

So....the intersection  of  these (in base 10 )  is  [  25, 63]

 

The average  of  these integers   = 

 

[ 25 + 63 ]  / 2  =  

 

44

 

 

cool cool cool

 Dec 24, 2020
 #3
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great job both of you!

 Dec 24, 2020

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