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 August 12th, 2011, 10:00 AM #1 Newbie   Joined: Aug 2011 Posts: 2 Thanks: 0 How to proof this is true by induction **STUCK** Hi guys, first of all thanks for your help. I'm stuck with this problem, i need to proof the following: 5^(n+1) + 2 * 3^n + 1 is divisible by 8 for any n >= 0 It's true for n = 0 5 + 2 + 1 = 8 How to proof that it's true for n+1 ?? Thanks!!
 August 12th, 2011, 10:20 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,204 Thanks: 511 Math Focus: Calculus/ODEs Re: How to proof this is true by induction **STUCK** I would set my induction hypothesis $P_n$ as: $5^{n+1}+2\cdot3^n+1=8m$ where $m\in\mathbb N$ Then compute: $$$5^{n+2}+2\cdot3^{n+1}+1$$-$$5^{n+1}+2\cdot3^n+1$$=5^{n+1}$$5-1$$+2\cdot3^{n}$$3-1$$=4$$5^{n+1}+3^n$$$ Now, we may write: $5^{n+1}+3^n=2k$ where $k\in\mathbb N$ since the sum of two odd numbers is even, thus adding equivalent expressions to both sides of the induction hypothesis, we have: $5^{n+1}+2\cdot3^n+1+$$5^{n+2}+2\cdot3^{n+1}+1$$-$$5^{n+1}+2\cdot3^n+1$$=8m+8k$ $5^{(n+1)+1}+2\cdot3^{n+1}+1=8$$m+k$$$ Since we have derived $P_{n+1}$ from $P_n$ this completes the proof by induction.
 August 12th, 2011, 10:52 AM #3 Newbie   Joined: Aug 2011 Posts: 2 Thanks: 0 Re: How to proof this is true by induction **STUCK** Thanks Mark!
 August 12th, 2011, 10:57 AM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,204 Thanks: 511 Math Focus: Calculus/ODEs Re: How to proof this is true by induction **STUCK** Glad to help, and welcome to the forum!

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