For integers x,y, notation xy means that x is a divisor of y. (For example 530, but it is not true that 521.) N denotes the set of natural numbers. Consider the following claim:

Claim: nN57ⁿ2ⁿ.

The inductive proof of this claim is given below, except that in the inductive step, the consecutive steps in the derivation are out of order. Give a correct ordering of the steps in this derivation.

Proof: We apply mathematical induction. In the base case, when n=0, we have 7⁰2⁰=11=0=50, so the claim holds for n=0.

In the inductive step, assume that the claim holds for n=k, that is 57^k2^k. This means that 7^k2^k=5b, for some integer b. We now proceed as follows:

(1) So 57^k+12^k+1, completing the inductive step.

(2) We have 7^k+12^k+1=77^k22^k

(3) So 7^k+12^k+1=5(7k+2b)

(4) So 7^k+12^k+1=57^k+2(7^k-2^k)

(5) So 7^k+12^k+1=57^k+2(5b)

We proved the base case, for n=0, and that the claim holding for n=k implies that it holds for n=k+1. By the principle of induction, this proves the claim for all n. QED

Choose the correct ordering in the derivation above:

1-2-3-4-5

2-5-4-3-1

2-4-5-3-1

3-2-1-4-5