Question 1

What happens if I can't prove the inductive step?

Accepted Answer

First, double-check your algebra. It's the most common source of error. Second, ensure you have correctly used the inductive hypothesis ($P(k)$). A proof of $P(k+1)$ that doesn't rely on $P(k)$ is not a proof by induction. If you're still stuck, reconsider your approach. For divisibility, is $f(k+1)-f(k)$ helpful, or maybe $f(k+1) - c \cdot f(k)$ for some constant $c$?

Question 2

Can I work backwards from the $n=k+1$ case?

Accepted Answer

You must be very careful. It is not valid to assume the $n=k+1$ result is true and work backwards to the $n=k$ assumption. However, you can use the target expression for $n=k+1$ as a 'goal' on your scrap paper to guide the manipulation of your starting expression. Your final written proof must flow logically from the LHS to the RHS of the $P(k+1)$ statement.

Question 3

Do I have to write the full concluding sentence every time?

Accepted Answer

Yes, absolutely. Examiners look for it. A statement like 'Since the statement is true for $n=1$ and if it's true for $n=k$ it is also true for $n=k+1$, it is true for all positive integers $n$ by the principle of mathematical induction' is essential for full marks. It shows you understand the logical principle you are using.

Question 4

Is proof by induction only for positive integers?

Accepted Answer

No. The statement might be for all integers $n \ge n_0$, where $n_0$ could be any integer, such as 0, 2, or even -5. Your base case would then be to prove the statement for $n=n_0$, and your assumption would be for $k \ge n_0$.

Proof by induction

What you need to know

The Domino Effect of Proofs