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Math 310: Proofs By Induction Worksheet – Partial Solutions
- Prove that for all n ≥ 4 , 3 n^ ≥ n^3.
Scratch work:
(a) What is the predicate P(n) that we aim to prove for all n ≥ n 0?
P (n) : 3n^ ≥ n^3
(b) What is n 0 =? = 4 (c) So the base case consists of proving P (n 0 ). Write out what this means, specifically for this problem.
P (4) : 3^4 ≥ 43
Now verify it. Since 34 = 81 and 43 = 64, clearly P (4) holds. (d) The induction step is to show that P (k) => P (k + 1) (for any k ≥ n 0 ). Spell this out.
3 k^ ≥ k^3 ⇒ 3 k+1^ ≥ (k + 1)^3 for any k ≥ 4
i. The Induction Hypothesis is P (k). Write it out.
P (k) : 3k^ ≥ k^3
ii. Write out the goal: P (k + 1).
P (k + 1) : 3k+1^ ≥ (k + 1)^3
iii. Rewrite the LHS of P (k + 1) until you can relate it to the LHS of P (k). 3 k+1^ = 3k^3 ˙ ≥ 3 k˙^3 iv. Rewrite the RHS of P (k + 1) until you can relate it to the RHS of P (k).
(k + 1)^3 = k^3 + 3k^2 + 3k + 1.
Want to show that this is less or equal to 3 k˙^3 v. The induction hypothesis gives you the inequality between certain ”chunks” of the RHS and LHS of P (k+1). It remains to compare the remaining parts and show that the inequality holds between those too. Can you think of a way? Use the back of the page to write a clear, correct, succinct proof of the statement.
——Try to figure this out for Wednesday————
- Prove that 7 divides 2 n+2^ + 3^2 n+1^ for any non-negative integer n.
Scratch work:
(a) What is the predicate P(n) that we aim to prove for all n ≥ n 0?
P (n) : 7 divides 2 n+2^ + 3^2 n+
(b) What is n 0 =? = 0 (c) So the base case consists of proving P (n 0 ). Write out what this means, specifically for this problem.
P (0) : 7 divides 2 0+2^ + 32(0)+
Now verify it. Since 2 0+2^ + 32(0)+1^ = 2^2 + 3 = 7, certainly 7 divides it. (d) The induction step is to show that P (k) => P (k + 1) (for any k ≥ n 0 ). Spell this out.
If 7 divides 2 k+2^ + 3^2 k+1^ for some k ≥ 0 , then it must also divide 2 k+3^ + 3^2 k+
i. The Induction Hypothesis is P (k). Write it out. P (k) : 2k+2^ + 3^2 k+1^ = 7a for some integer a
ii. Write out the goal: P (k + 1). P (k + 1) : 2k+3^ + 3^2 k+3^ = 7b for some integer b
iii. Rewrite the LHS of P (k + 1) until you can relate it to the LHS of P (k).
2 k+3+3^2 k+3^ = 2k+22+3^2 k+19 = 2k+22+3^2 k+1(2+7) = 2(2k+2+3^2 k+1)+3^2 k+1 7.
iv. Prove the induction step entirely.
By induction hypothesis, 2 k+2^ + 3^2 k+1^ = 7a, so 2 k+3^ + 3^2 k+3^ = 2(7a) + 3^2 k+17 = 7(2a + 3k+1).
Use the back of the page to write a clear, correct, succint proof of the statement.
