SOLUTION: using mathematicsl induction prove that 3^n+7^n-2 is divisible by 8

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Question 982823: using mathematicsl induction prove that 3^n+7^n-2 is divisible by 8
Answer by Edwin McCravy(20056) About Me  (Show Source):
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using mathematicsl induction prove that 3%5En%2B7%5En-2 is divisible by 8
f%28n%29+=+3%5En%2B7%5En-2
f%281%29+=+3%5E1%2B7%5E1-2=3%2B7-2=8 is divisible by 8.

We only need to show that the difference f(n+1)-f(n) is divisible by 8. 
For if that difference is divisible by 8 then the assumption that f(n) 
is divisible by 8 will show that f(n+1) is divisible by 8 because the 
sum of the difference and f(n) will be f(n+1) and the sum of two 
multiples of 8 is a multiple of 8. 

f%28n%2B1%29-f%28n%29+=+%283%5E%28n%2B1%29%2B7%5E%28n%2B1%29-2%29-%283%5En%2B7%5En-2%29
f%28n%2B1%29-f%28n%29+=+3%5E%28n%2B1%29%2B7%5E%28n%2B1%29-2-3%5En-7%5En%2B2%29
f%28n%2B1%29-f%28n%29+=+3%5E%28n%2B1%29-3%5En%2B7%5E%28n%2B1%29-7%5En}
f%28n%2B1%29-f%28n%29+=+3%5En%2A%283-1%29%2B7%5En%2A%287-1%29
f%28n%2B1%29-f%28n%29+=+3%5En%2A%282%29%2B7%5En%2A%286%29
f%28n%2B1%29-f%28n%29+=+2%2A3%5En%2B6%2A7%5En
f%28n%2B1%29-f%28n%29+=+2%283%5En%2B3%2A7%5En%29
f%28n%2B1%29-f%28n%29+=+2%283%283%5E%28n-1%29%2B7%5En%29%5E%22%22%29

But we must now show that 

g%28n%29=3%5E%28n-1%29%2B7%5En is divisible by 4 
g%281%29=3%5E%281-1%29%2B7%5E1=3%5E0%2B7=1%2B7=8=4%2A2 is divisible by 4

As above, we only need to show that the difference g(n+1)-g(n) is divisible by
4. For if that is divisible by 4 then the assumption that g(n) is divisible
by 4 will show that g(n+1) is divisible by 4 because the sum of two
multiples of 4 is a multiple of 4.


g%28n%2B1%29-g%28n%29=3%5E%28n%2B1-1%29%2B7%5E%28n-1%29-3%5E%28n-1%29-7%5En
g%28n%2B1%29-g%28n%29=3%5En%2B7%5E%28n-1%29-3%5E%28n-1%29-7%5En
g%28n%2B1%29-g%28n%29=3%5En-3%5E%28n-1%29%2B7%5E%28n-1%29-7%5En
g%28n%2B1%29-g%28n%29=3%5E%28n-1%29%2A%283-1%29%2B7%5E%28n-1%29%281-7%29
g%28n%2B1%29-g%28n%29=3%5E%28n-1%29%2A%282%29%2B7%5E%28n-1%29%28-6%29
g%28n%2B1%29-g%28n%29=2%2A3%5E%28n-1%29-6%2A7%5E%28n-1%29
g%28n%2B1%29-g%28n%29=2%283%5E%28n-1%29-3%2A7%5E%28n-1%29%29

The expression in parentheses is even, because
all products and powers of odd numbers are odd numbers,
and the difference of two odd numbers is an even number.
Therefore g%28n%2B1%29-g%28n%29 is a multiple of 4.

Therefore f%28n%2B1%29-f%28n%29 is a multiple of 8.

Edwin