I started to use algebra to work with this (using n+1), but I have no idea where exactly to go with it. Assistance would be greatly appreciated.
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True for n=1, 1/3+1/5+7/15=1 which is an integer
True for n
(5n^3 + 3n^5 + 7n)/15 so this is an integer
Put n+1 th term, forget it is all over 15 for the moment
3(n+1)^5 +5(n+1)^3+7(n+1) =
3(n^5+5n^4 +10n^3+10n^2+5n+1)+
5(n^3+3n^2+3n+1) +7n+7
Adding up
(3n^5+15n^4 +30n^3+30n^2+15n+3)+
(5n^3+15n^2+15n+5) +7n+7=
(3n^5+15n^4 +35n^3+45n^2+37n+15)
Know
( 5n^3 + 3n^5 + 7n) is a multiple of 15
Take away
So now want to know about 15n^4+30n^3+45n^2+30n +15
Each term is a multiple of 15 so when divide by 15 we get an integer
So by induction true for 1, and if true for n then it is for n+1
so expression is always an integer
True for n
(5n^3 + 3n^5 + 7n)/15 so this is an integer
Put n+1 th term, forget it is all over 15 for the moment
3(n+1)^5 +5(n+1)^3+7(n+1) =
3(n^5+5n^4 +10n^3+10n^2+5n+1)+
5(n^3+3n^2+3n+1) +7n+7
Adding up
(3n^5+15n^4 +30n^3+30n^2+15n+3)+
(5n^3+15n^2+15n+5) +7n+7=
(3n^5+15n^4 +35n^3+45n^2+37n+15)
Know
( 5n^3 + 3n^5 + 7n) is a multiple of 15
Take away
So now want to know about 15n^4+30n^3+45n^2+30n +15
Each term is a multiple of 15 so when divide by 15 we get an integer
So by induction true for 1, and if true for n then it is for n+1
so expression is always an integer