My Math Forum Using Mathematical Induction

 Calculus Calculus Math Forum

 February 19th, 2018, 12:34 PM #1 Senior Member   Joined: May 2014 From: Allentown PA USA Posts: 110 Thanks: 6 Math Focus: dynamical systen theory Using Mathematical Induction Dear My Math Forum Community: Prove: n^3 - n is divisible by 3 for every positive integer n. [ Hint: use (k + 1)^3 - (k + 1) = (k^3 - k) + 3(k^2 + k). This proof is to illustrate the concept of mathematical induction. ]
 February 19th, 2018, 12:51 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 535 Thanks: 306 Math Focus: Dynamical systems, analytic function theory, numerics Base case is $n=1$ and 0 is clearly divisible by 3 so it holds. Now assume inductively that it holds for $k^3 - k$. Applying the hint you have $(k + 1)^3 - (k + 1) = (k^3 - k) + 3(k^2 + k)$ so by your inductive assumption, the left summand is a multiple of 3. Clearly the right summand is as well and therefore, $(k+1)^3 - (k+1)$ is a multiple of which and by induction, it holds for all $k \in \mathbb{N}$.
February 19th, 2018, 12:51 PM   #3
Senior Member

Joined: Aug 2012

Posts: 2,135
Thanks: 621

Quote:
 Originally Posted by Carl James Mesaros Dear My Math Forum Community: Prove: n^3 - n is divisible by 3 for every positive integer n.
Yet another problem better solved without induction.

$n^3 - n = n (n^2 -1) = n (n + 1) (n - 1)$

One of $n$, $n + 1$, and $n -1$ is divisible by $3$; therefore their product is.

February 19th, 2018, 12:54 PM   #4
Senior Member

Joined: Sep 2016
From: USA

Posts: 535
Thanks: 306

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
 Originally Posted by Maschke Yet another problem better solved without induction. $n^3 - n = n (n^2 -1) = n (n + 1) (n - 1)$ One of $n$, $n + 1$, and $n -1$ is divisible by $3$; therefore their product is.
I think the point is not to prove this "theorem" which is trivial. Rather, it is to learn how to apply induction, which the OP failed to do since he just asked someone to do it for him.

 February 19th, 2018, 12:54 PM #5 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,900 Thanks: 1094 Math Focus: Elementary mathematics and beyond $n^3-n=n(n-1)(n+1)=(n-1)n(n+1).$ Now it's easily seen that this is divisible by 3. If you need to solve this by induction, you can start by telling us what your initial approach would be.
 February 20th, 2018, 10:45 AM #6 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 If you are required to use induction, then you start, of course, by observing that when n= 1, this is 1- 1= 0 which is divisible by 6. The next step is to show that "if, for some positive integer k, k^3- k is divisible by 6 then so is (k+1)^3- (k+1)". To do that, calculate (k+1)^3- (k+1)= k^3+ 3k^2+ 3k+ 1- k- 1= k^3+ 3k^2+ 2k= (k^3- k)+ (3k^2+ 3k). Since k^3- k was divisible by 6 we can write this as 6n+ 3k(k+ 1) and we now can observe that, since k and k+ 1 are consecutive integers one of them is divisible by 2.

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Lambert1 Abstract Algebra 3 November 7th, 2015 06:33 AM creatingembla Calculus 3 October 19th, 2015 10:46 AM alloy Algebra 1 December 18th, 2012 08:30 AM alloy Algebra 3 November 14th, 2012 04:54 AM Sunde Algebra 14 August 3rd, 2011 05:10 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top