prove by induction that 4^n-1 is divisible by three for all posotive integers of n
any help would be appriciated thanks.
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Initialization of induction is easy.
Let's suppose that for a given n is divisible by 3.
You need to demonstrate that is divisible by 3.
Hint : .
prove is divisible by 3
assume true for , show true for ...
since is divisible by 3, then , a positive integer.
since is also a positive integer, is a multiple of 3, and therefore, is divisible by 3.
Here's another way . . . Prove by induction that for all positive integers
Verify . . . true
Add to both sides: .
Factor: . . . . . . . . . . . . . . . . .
We have proved . . . The inductive proof is complete.
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