Solving Problem: Prove 3^{2n+2} - 8n -9 Divisible by 64

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Discussion Overview

The discussion revolves around proving that the expression 3^{2n+2} - 8n - 9 is divisible by 64 for any positive integer n. The scope includes mathematical reasoning and problem-solving techniques, particularly focusing on induction and algebraic manipulation.

Discussion Character

  • Mathematical reasoning, Homework-related, Exploratory

Main Points Raised

  • One participant suggests that induction is a common method for proving properties of natural numbers.
  • Another participant proposes examining the binomial expansion of 9^n in relation to the expression, considering the relationship between 9 and 8.
  • A different participant discusses the algebraic manipulation of expressions, indicating a potential method for proving divisibility.
  • One participant clarifies that proving one expression's divisibility by 64 can imply the divisibility of another related expression, with a note on the case when n=0.
  • A later reply indicates understanding of the previous points raised in the discussion.

Areas of Agreement / Disagreement

Participants explore various approaches to the problem, but no consensus is reached on a definitive method or solution. Multiple competing views and techniques remain present in the discussion.

Contextual Notes

Some assumptions about the properties of the expressions and the conditions under which they hold may be missing or unresolved, particularly regarding the case when n=0.

recon
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Can someone point me in the right direction of solving the following problem:

Prove that for any positive integer n, the value of the expression [tex]3^{2n+2} - 8n -9[/tex] is divisible by 64.
 
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I find when you're asked to prove that every member of some subset of the natural numbers has some property, induction usually works.
 
9^{n+1}-8n-9

or equivalently

9^n-8n-1

what is the binomial expansion of 9^n when considering 9=8+1?
 
Maybe I'm having a blonde moment, but doesn't [itex]a^{n + 1} - a = a(a^{n} - 1)[/itex], making [itex]9^{n + 1} - 9 - 8n = 9(9^{n} - 1) - 8n[/itex]?
 
ah, perhaps i ought to have been clearer: i wasn't say the epxressions are equal, but that if you prove one is divisible by 64 for all n, the other will be divislbe by 64 for all n (give or take a case when n=0). I let m=n+1 in the first, then relabelled n=m.
 
I get it now. Thank you.
 

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