My Math Forum Proof by Induction - Divisibility Proofs

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September 19th, 2011, 09:59 AM   #1
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Proof by Induction - Divisibility Proofs

Q. Prove by induction that... (see attachment)

Notes: For the line beginning "Show that n = k + 1..." I think I may have gone wrong in the calculation here, but I'm not exactly sure.
Thank you.
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 September 19th, 2011, 10:19 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs Re: Proof by Induction - Divisibility Proofs We need to prove by induction: $7^{2n+1}+1=8m$ where $n,m\in\mathbb N$ You have already shown the base case $P_1$ is valid. Thus, our induction hypothesis $P_k$ is: $7^{2k+1}+1=8m$ Let's add a multiple of 8 to each side: $$$7^2-1$$7^{2k+1}=8p$ $7^{2k+1}+1+$$7^2-1$$7^{2k+1}=8m+8p$ $7^{2(k+1)+1}+1=8(m+p)$ We have derived $P_{k+1}$ from $P_k$ completing the proof by induction.
 September 19th, 2011, 11:20 AM #3 Senior Member   Joined: Sep 2011 Posts: 395 Thanks: 0 Re: Proof by Induction - Divisibility Proofs Thank you for the help. Much obliged.

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