As i understand it should be 3n+4n+5n
To prove that number is divisible by 12 we will prove that it is divisible by both 3 and 4
Divisible by 3:
3n is obviously divisible by 3 for every odd integer n, then the point is to prove that 4n+5n is divisible by 3
According to abbreviated multiplication formula for odd degrees:
4n+5n=(4+5)(4n−1−4n−2∗5+4n−3∗52−...−4∗5n−2+5n−1)=9∗(4n−1−4n−2∗5+4n−3∗52−...−4∗5n−2+5n−1)⋮3
Divisible by 4:
4n is obviously divisible by 4 for every odd integer n, then the point is to prove that 3n+5n is divisible by 4
According to abbreviated multiplication formula for odd degrees:
3n+5n=(3+5)(3n−1−3n−2∗5+3n−3∗52−...−3∗5n−2+5n−1)=8∗(3n−1−3n−2∗5+3n−3∗52−...−3∗5n−2+5n−1)⋮4
The statement has been proven
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