# Complete Induction proof

• October 21st 2009, 06:46 AM
coobe
Complete Induction proof
can someone give me a hint on how to solve such questions in general ?

"Proof with complete induction that every number with $2^{3n}-1$ with n element of Natural numbers, is divisible by 7"

• October 21st 2009, 07:14 AM
tonio
Quote:

Originally Posted by coobe
can someone give me a hint on how to solve such questions in general ?

"Proof with complete induction that every number with $2^{3n}-1$ with n element of Natural numbers, is divisible by 7"

$\mbox{For}\,n=1\,\,\mbox{we get}\,2^{3\cdot1}-1=8-1=7\,\mbox{ so it is true}$

$\mbox{Assume now for }\,\,n\,\mbox{ and we'll show for }\,\,n+1:$

$2^{3(n+1)}-1=2^{3n+3}-1=8\cdot2^{3n}-1=2^{3n}-1+7\cdot2^{3n}$

Apply now the inductive hypothesis and we're through.

Tonio
• October 22nd 2009, 06:41 AM
CaptainBlack
Quote:

Originally Posted by coobe
can someone give me a hint on how to solve such questions in general ?

"Proof with complete induction that every number with $2^{3n}-1$ with n element of Natural numbers, is divisible by 7"

Complete (or mathematical) induction is a method of proving an assertion $P(n)$ is true for every natural number by first showing that the base case (that is $P(1)$ or $P(0)$ ) is true, then showing that the truth of $P(k)$ for some $k$ implies the truth of $P(k+1)$. When we have done these two things we have proven $P(n)$ for all $n \in \mathbb{N}$.