can someone give me a hint on how to solve such questions in general ?
"Proof with complete induction that every number with $\displaystyle 2^{3n}-1$ with n element of Natural numbers, is divisible by 7"
thanks in advance
$\displaystyle \mbox{For}\,n=1\,\,\mbox{we get}\,2^{3\cdot1}-1=8-1=7\,\mbox{ so it is true}$
$\displaystyle \mbox{Assume now for }\,\,n\,\mbox{ and we'll show for }\,\,n+1:$
$\displaystyle 2^{3(n+1)}-1=2^{3n+3}-1=8\cdot2^{3n}-1=2^{3n}-1+7\cdot2^{3n}$
Apply now the inductive hypothesis and we're through.
Tonio
Complete (or mathematical) induction is a method of proving an assertion $\displaystyle P(n)$ is true for every natural number by first showing that the base case (that is $\displaystyle P(1)$ or $\displaystyle P(0)$ ) is true, then showing that the truth of $\displaystyle P(k)$ for some $\displaystyle k$ implies the truth of $\displaystyle P(k+1)$. When we have done these two things we have proven $\displaystyle P(n)$ for all $\displaystyle n \in \mathbb{N}$.
Now look at Tonio's solution and you will see that it fits this proof method.
Complete induction is often taken as an axiom of arithmetic.
CB