Order 8 of the following sentences so that they form a proof by induction of the statement: 2^𝑛>4𝑛 for 𝑛>4.

Question

Order 8 of the following sentences so that they form a proof by induction of the statement: 2ⁿ>4n for 𝑛>4.

Options:

1. Base Case: n=5
2. Inductive Hypothesis: Assume 2^k>4^k
3. LHS: 2⁵=32, RHS: 4×5=20, so 2⁵>4×5
4. Inductive Hypothesis: Assume 2ⁿ>4n for all natural numbers n
5. We will proceed by induction to prove that 2ⁿ>4n for n>4
6. Inductive Hypothesis: Assume 2^k>4k for some natural number k
7. Base Case: n=4
8. LHS 2⁴=16, RHS: 4×4=16 so 2⁴≥4×4
9. 2^k+1=2^k×2>2×4k=8k
10. And, 8k=4k+4k>4k+4=4(k+1)
11. So, 2^k+1>4(k+1)
12. Then by mathematical induction, 2ⁿ>4n for n>4

Answer 1

4, 1, 3, 6, 9, 10,11, 12