Proof by induction — common mistakes

In the inductive step, you MUST explicitly use your assumption for $n=k$. A good habit is to underline or highlight where you substitute it in your own working to ensure you haven't missed it. Also, never skip the final concluding sentence; it's usually worth a specific mark and ties the whole proof together.

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 $c$?

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.

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.

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 $k \ge n_0$.