(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove the statement by mathematical induction:

5^{n}+ 9 < 6^{n}for all integers n≥2

2. Relevant equations

..

3. The attempt at a solution

Proof: let P(n) be the statement,

5^{n}+ 9 < 6^{n}

P(2) is true because,

34<36.

Suppose that P(n) is true.

P(n+1) would be,

5^{n+1}+ 9 < 6^{n+1}

6^{n+1}= 6[itex]\bullet[/itex]6^{n}

6[itex]\bullet[/itex](5^{n}+9)< 6[itex]\bullet[/itex]6^{n}

==> 54+ 6(5^{n}) < 6[itex]\bullet[/itex]6^{n}

==> 54+ 5(5^{n}) + 5^{n}< 6[itex]\bullet[/itex]6^{n}

==> 54+ 5^{n+1}+ 5^{n}< 6[itex]\bullet[/itex]6^{n}

What do I do from here? I don't know how to wrap up and prove this induction, or that P(n+1) is real? Any help would be much appreciated!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Proof by induction: 5^n + 9 < 6^n for all integers n≥2

**Physics Forums | Science Articles, Homework Help, Discussion**