Is 11^n - 4^n a Multiple of 7? Proving with Mathematical Induction

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SUMMARY

The discussion centers on proving that the expression 11n - 4n is a multiple of 7 using mathematical induction. The initial step involves substituting k+1 for n, leading to the expression 11(11k) - 4(4k). The key insight is recognizing that 11 can be expressed as 7 + 4, which simplifies the proof process. Participants emphasize the importance of correctly applying the induction hypothesis to complete the proof.

PREREQUISITES
  • Understanding of mathematical induction
  • Familiarity with algebraic manipulation
  • Basic knowledge of modular arithmetic
  • Experience with sequences and series
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  • Study the principles of mathematical induction in detail
  • Learn about modular arithmetic and its applications
  • Practice algebraic manipulation techniques for proofs
  • Explore examples of induction proofs involving sequences
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Students in mathematics, educators teaching algebra and number theory, and anyone interested in learning proof techniques through mathematical induction.

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Homework Statement


Prove that
[tex] 11^n - 4^n[/tex]
is a multiple of 7

Homework Equations


N/A

The Attempt at a Solution


I substituted k+1 in for n and simplified to get
[tex] 11(11^k)-4(4^k)[/tex]
but after this point I get stuck. Any help would be appreciated.
 
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Use that 11=7+4.
 

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