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.
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
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