Proof by Induction: Divisibility by 17

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    Induction Proof
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Discussion Overview

The discussion revolves around proving the divisibility of the expression \(3 \cdot 5^{2n+1} + 2^{3n+1}\) by 17 using mathematical induction. Participants explore the base case and inductive step, addressing issues in the formulation and clarity of the expression.

Discussion Character

  • Homework-related, Mathematical reasoning, Exploratory

Main Points Raised

  • One participant presents the expression to be proven and requests assistance.
  • Another participant points out a typographical error in the expression, indicating a misplaced parenthesis that affects clarity.
  • A different participant suggests that the original poster should attempt the proof themselves and share where they encounter difficulties.
  • A participant describes their progress in proving the base case for \(n=1\) and outlines their approach for the inductive step, noting they encountered challenges in continuing the proof.
  • Later, a participant claims to have resolved their confusion and reformulates the expression in terms of \(K\) and \(L\), suggesting a method to show that the sum of two numbers divisible by 17 is also divisible by 17.

Areas of Agreement / Disagreement

The discussion includes multiple perspectives on how to approach the proof, with no consensus reached on the correctness of the proposed methods or solutions. Participants express varying levels of understanding and progress.

Contextual Notes

There are unresolved issues regarding the clarity of the original expression due to a typographical error, as well as potential gaps in the inductive proof steps that have not been fully addressed.

guropalica
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Proof by induction that 3 * 5^2n+1) + 2^3n+1 is divisible by 17!
Thanks in advance guys
 
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In your expression, there is an ")" without an "(" --> 3 * 5^2n+1) + 2^3n+1

so it is not clear what do to!
 
You also need to try yourself and let us know where you're stuck. Then we can help.
 
I proved the base case n=1, and then I try doing the step case assuming that it satisfies for any k, then I try proving it by k+1.
I got sth like 24 * (3 * 5^(2k+1)) + 7 * 2^(3k +1) Now I'm stuck can't continue, I don't have any ideas :/
btw the initial equation is 3 * 5^(2n+1) + 2^(3n+1) !
 
Got it now :) let's denote the initial statement as K + L, so we have sth like 24l + 7k = 7(k+l) + 17k, sum of two numbers divisible by 17 is divisible by 17, anyway thx
 

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