# Mathematical induction exercise

So I am having trouble with an exercise about mathematical induction. I have the following sentence: $$1^{n+1}$$ < $$2^n$$ for every n ≥ 3

Now, what I would personally do is:

First prove that it is true for n = 3

$$1^{3+1}$$ = 1 < 8 = $$2^3$$

And assume that if the sentence is true for n, then it is also true for k. Then I would prove that the sentence is true for k+1 for every k ≥ 3.

Now the problem is that I have seen an answer to a question similar to this, where the person solving the problem proved that the sentence is true for k+1 for every k ≥ 4.

Even when that person changed k ≥ 3 to k ≥ 4, it didn't make any change to the overall proof. What I want to know is, which notation is the right one; k ≥ 3, or k ≥ 4?

• IMHO, this exercise is quite meaningless, for there’s almost nothing to prove by induction here: in fact, $2^n > 2^0 = 1 = 1^{n+1}$ for every $n \in \mathbb{N} \setminus \{0\}$. Commented Oct 12, 2019 at 10:18

It is a strange identity to prove by induction since it is trivially true!

Anyway for the induction step we assume true

$$1^{k+1} < 2^k$$

and we need to prove that

$$1^{k+2} < 2^{k+1}$$

which is true indeed

$$1^{k+2} = 1\cdot 1^{k+1}< 2^k <2^{k+1}$$

Since the base case has been proved for $$n=3$$, the last needs to be true for $$k\ge 3$$.

It should be $$k\ge 3$$. The other person's proof only works if you check $$n=4$$ as part of the base step. (It's a strange exercise, mind, because even starting at $$n=0$$ would work.)

• Would it be right to check n=4 as part of the base step then? Shouldn't you always check for the statement in the original question (in this case n =< 3)? Commented Oct 12, 2019 at 10:17
• @Setin If you want a proof the result is true for $n\ge3$, but you only check going from $n=k$ to $n=k+1$ works for $k\ge4$, your base step needs to include $n=4$ as well as $n=3$. Ideally, one should note the inductive-step proof works for $k\ge0$, so as to prove the result for $n\ge0$ using only $n=0$ as the base step.
– J.G.
Commented Oct 12, 2019 at 10:19