Study case: proving a summation formula by induction In this A-Level Proof by Induction Problem Tutor, I use one classic study case to make proof feel structured rather than mysterious. The problem is: prove that 1 + 2 + 3 + ... + n = n(n + 1)/2 for all positive integers n. I like this example because students can already test the formula with small numbers, but they still need a formal method to prove it for every positive integer. The goal is not just to copy an induction template. The goal is to understand why the base case, assumption and induction step work together. Before solving, I ask the student to test the statement for a few values. If n = 1, the left side is 1 and the right side is 1(2)/2 = 1. If n = 3, the left side is 1 + 2 + 3 = 6 and the right side is 3(4)/2 = 6. This testing does not prove the result for all n, but it builds confidence that the statement is reasonable. I then explain that proof by induction is like a line of dominoes: show the first one falls, then show that any falling domino forces the next one to fall. Method 1 is the standard induction structure. First, I prove the base case. For n = 1, the left side is 1 and the right side is 1(1 + 1)/2 = 1. Therefore the statement is true for n = 1. I write this clearly because a missing base case makes the proof incomplete. The base case anchors the chain. Next, I make the induction assumption. I assume the result is true for n = k. That means 1 + 2 + 3 + ... + k = k(k + 1)/2. I remind students that we are not assuming the whole result is true forever. We are assuming one case, the k case, so that we can prove the next case. This distinction matters because induction is often misunderstood as circular reasoning. Then I prove the result for n = k + 1. The left side becomes 1 + 2 + 3 + ... + k + (k + 1). Using the assumption, the first part 1 + 2 + ... + k can be replaced by k(k + 1)/2. So the expression becomes k(k + 1)/2 + (k + 1). I factor out (k + 1), giving (k + 1)(k/2 + 1). This is (k + 1)(k + 2)/2, which is exactly the formula with n = k + 1. Therefore the statement is true for k + 1 if it is true for k. Method 2 is the target-shape method. I ask the student to write down what they are trying to reach before manipulating the algebra. For n = k + 1, the right side should be (k + 1)((k + 1) + 1)/2, which simplifies to (k + 1)(k + 2)/2. When students know the target, the algebra feels less random. They can see why factoring out (k + 1) is a useful step. Method 3 is the visual pairing explanation. I show that the sum 1 + 2 + ... + n can be paired from the ends: 1 with n, 2 with n - 1, and so on. This gives pairs that each total n + 1. This is not the induction proof, but it helps the student understand why the formula is true. I use this as a conceptual support alongside the formal proof. Students often write induction better when the formula itself has meaning. The most common mistakes are forgetting the base case, writing assume true for n without specifying n = k, failing to use the induction assumption, and stopping before showing the exact k + 1 form. I also see students write a final sentence too vaguely. A strong proof needs a clear conclusion: since the result is true for n = 1 and true for k + 1 whenever it is true for k, it is true for all positive integers n by induction. This topic is important because proof questions test reasoning, not only calculation. I encourage students to slow down and use precise language. Phrases such as assume true for n = k and therefore true for n = k + 1 are not decoration; they are the logical bridge in the proof. If the writing is careless, the proof can become hard to follow even when the algebra is correct. A useful extension is to prove a divisibility result, such as 3^n - 1 is divisible by 2 for positive integers n. Another extension is to prove a formula involving squares or cubes. These examples help students see that induction is a general proof method, not only a technique for sums. I introduce extensions once the student is comfortable with the standard structure. In one-to-one tutoring, I would finish by giving a similar proof with a scaffold, then gradually remove the scaffold. I want the student to be able to name the three parts of the proof and explain why each one is needed. With clear practice, proof by induction becomes a reliable routine rather than a frightening A-Level topic. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit A-Level Maths tutoring: https://www.mastermathstutoring.co.uk/a-level-maths. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.