- #1
McFluffy
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- 1
Homework Statement
Prove ##5^n+9<6^n## for ##n\epsilon \mathbb{N}|n\ge2## by induction.
Homework Equations
None
The Attempt at a Solution
The base case which is when ##n=2##:
##5^2+9<6^2##
##34<36##
Thus, the base case is true. Now for the induction step.
Induction hypothesis: Assume ##5^k+9<6^k## for ##k \epsilon \mathbb{N}|k\ge2##
Induction step: To show that ##5^k+9<6^k## implies ##5^{k+1}+9<6^{k+1}##, I'll start with the induction hypothesis:
##5^k+9<6^k##
##5(5^k+9)<6(6^k)## since we have an inequality and not an equation, I can add numbers that are bigger to one side and not violate inequality. With this in mind, since ##5<6##, I'd multiplied 5 on the smaller side and 6 on the bigger side.
##5^{k+1}+45<6^{k+1}##
##5^{k+1}+9<6^{k+1}##
Multiply one side by a bigger number than the other side is what makes this proof possible. And I think this is a logical thing to do because we're dealing with an inequality and not an equation. Are my steps justified?