So I have to prove that $3^n$ is greater than or equal to $3n$ using induction. The base case is a not a problem, but I can't seem to figure out where to go for $(n-1)$. I've tried saying:
$$3^n=3\cdot3^{n-1}>3\cdot3(n-1)$$
$$3\cdot3(n-1)=9n-9$$
I'm pretty sure my end goal is $3n$, but I'm not really sure how to get there.
Any suggestions would be much appreciated.
Answer
I will show that we may assume that the inequality holds for some $k$ and use that to show that it holds for $k+1$.
Use the base case $n=2$, $3^2>3(2)$, which is obviously true.
Now, assume that for $n=k$ that $3^k>3k$. This is called the induction hypothesis. Now, we must prove the inequality for $k+1$.
$3^k>3k$ via our induction hypothesis.
$3\cdot3^k>3\cdot3k$ multiplying by $3$ on both sides.
$3^{k+1}>3\cdot3k>3k+3=3(k+1)$
Thus, the inductive step and our proof are complete.
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