Assessment: Mathematical Induction

What are the two required steps of a mathematical induction proof?

When proving a summation formula, what should be added to both sides during the inductive step?

Using 1+2+...+n = n(n+1)/2, find the sum of the first 20 positive integers.

Verify the base case for "5ⁿ − 1 is divisible by 4" at n=1: what is 5¹ − 1?

Verify the base case for "n! > 3ⁿ for n≥7" at n=7: what is 7!?

If P(5) is true and the inductive step "P(k) ⟹ P(k+1)" holds for all k ≥ 5, what is the smallest n > 5 for which P(n) is guaranteed true by induction?

A network engineer wants to prove a summation formula for the total cables needed to connect n junction boxes is correct for EVERY possible n, not just the sizes tested. Which technique proves this?

A cryptography engineer needs a computed value to always be divisible by a fixed number, at every step of an unbounded process. Which technique proves this holds at every step?

Why is the base case alone not sufficient to prove a statement for all positive integers?

The inequality "n² ≥ 2n+1" is false at n=1 (1≥3 is false) and n=2 (4≥5 is false), but becomes true at n=3 (9≥7 is true) and stays true afterward. What is the correct base case value to start this induction proof?