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One of the skills that A Level Maths students develop is the ability to prove statements about integers using mathematical reasoning and induction. A classic type of question is to show that a given algebraic expression is always divisible by a certain number. Let’s look at a typical example and walk through the steps.
The Problem
We are asked to prove that, for all positive integers n
n^3 + 3n^2 + 2n
is divisible by 6.
Step 1: Factorisation and First Observations
Let’s start by factoring the expression:
n^3 + 3n^2 + 2n = n(n^2 + 3n + 2)
Notice that n^2 + 3n + 2 can be factored further:
n^2 + 3n + 2 = (n + 1)(n + 2)
So, the expression becomes:
n(n + 1)(n + 2)
This is the product of three consecutive integers.
Step 2: Divisibility by 2 and 3
Let’s reason about divisibility:
- Divisibility by 2:
Among any three consecutive integers, one must be even, so their product is always divisible by 2. - Divisibility by 3:
Similarly, among any three consecutive integers, one of them must be a multiple of 3, so their product is also divisible by 3.
Since the product is always divisible by both 2 and 3, it must be divisible by 2 \times 3 = 6.
Step 3: Conclusion
Therefore, for any positive integer n, n^3 + 3n^2 + 2n is divisible by 6.
Why This Matters
Questions like this not only test your algebraic manipulation skills, but also your ability to apply logical reasoning about numbers. They frequently appear in exams and are a great way to strengthen your understanding of number theory. If you want to develop mastery in topics like this and get lots of practice with proofs and divisibility, an A Level Maths Revision Course can provide you with expert guidance and focused revision.
