Before any algebra, ask three questions: what must the last line say, what general form covers every case, and how do I get from one to the other? Following a finished proof is reading. Starting one is planning, and planning has its own method.
This page gives that method with one original example, so the first line stops being a blank.
Why can I follow but not start?
A model solution shows only the finished proof, not the planning behind it. You see n and n + 1 appear and assume the author simply knew to use them. The author chose them by looking at the target first.
So the skill you are missing is not algebra. It is working backwards from the final sentence to decide what to write first.
What are the three questions?
- Target. Write exactly what you have to show, in words and symbols. “Show that the result is odd” means the last line must read 2 × (something) + 1.
- General form. Pick letters that cover every case. “Two consecutive integers” becomes n and n + 1. “An odd number” becomes 2k + 1.
- Bridge. Do the algebra, then rewrite the result so it has the shape from question 1, and say why.
Worked example
Show that the sum of the squares of two consecutive integers is always odd.
Target. The final expression must have the form 2 × (integer) + 1.
General form. Let the integers be n and n + 1, where n is any integer.
Bridge. The sum of squares is n² + (n + 1)² = n² + n² + 2n + 1 = 2n² + 2n + 1.
Factor out 2 from the first two terms: 2n² + 2n + 1 = 2(n² + n) + 1.
Since n is an integer, n² + n is an integer, so this is 2 × (integer) + 1, which is odd. The statement is proved for every integer n.
Check on numbers (not a proof): n = 3 gives 9 + 16 = 25, and 2(9 + 3) + 1 = 25. n = −2 gives 4 + 1 = 5, and 2(4 − 2) + 1 = 5. Both agree.
What mistake do students make?
A student tries 3 and 4, gets 25, sees it is odd, tries 5 and 6, gets 61, and writes “so it is always odd”. It feels convincing.
It goes wrong because finitely many cases never cover every integer.
A statement can work for a hundred numbers and fail on the next one. The correction is to replace the numbers with n and let the algebra carry all cases at once, as above. Use numbers only as a final check, or to look for a pattern before proving it.
Another way to see the start
Sometimes the expression itself is the clue.
If you must show a quantity is a multiple of 4, look for a 4 to factor. If you must show it is positive for every real x, look for a completed square, because a square plus a positive constant cannot be negative. You can explore this shape in the quadratic structure explorer.
Self-check
- Show that n² + n is even for every integer n.
- Show that (2n + 1)² − 1 is a multiple of 8 for every integer n.
- Show that the sum of three consecutive integers is a multiple of 3.
Show answer
- n² + n = n(n + 1). One of two consecutive integers is even, so the product is even. (If n is even, n is the even factor. If n is odd, n + 1 is.)
- (2n + 1)² − 1 = 4n² + 4n + 1 − 1 = 4n² + 4n = 4n(n + 1). From question 1, n(n + 1) is even, so it equals 2m for some integer m. Then the expression is 4 × 2m = 8m, a multiple of 8.
- Take n − 1, n and n + 1. Their sum is (n − 1) + n + (n + 1) = 3n, which is a multiple of 3 because n is an integer.
How do I practise the habit?
For each proof, write the target and the general form on the first two lines before any algebra. Then let the working lead to the target’s shape. Keep your arithmetic tidy with the non-calculator working trainer when fractions or surds are involved.
Proof ideas appear in the vector work of two-dimensional vector proofs, for example showing points are collinear, and in advanced non-calculator reasoning, especially the lesson on a proof step a calculator cannot supply. The Additional Mathematics original practice hub collects questions to try.
If you still stall at the first line, online one-to-one Additional Mathematics tuition gives you an assigned teacher. They ask the planning questions as you work, then step back as you learn to ask them yourself.