Mathematics · Classes 9-10 · Chapter 1

Proofs with Consecutive Natural Numbers.

1·2·3·4 + 1 = 25, and 25 = 5². 2·3·4·5 + 1 = 121, and 121 = 11². Two perfect squares in a row. Luck? A few lines of algebra show it happens for any four consecutive numbers. This note has that trick, and a handful of short proofs about even and odd numbers.

1. Four in a row, plus one

Consecutive means one after another. Let four consecutive natural numbers be x, x + 1, x + 2, x + 3. One letter now stands for all four, and that is the first move of the proof.

Example 2 of the book shows that x(x + 1)(x + 2)(x + 3) + 1 is a perfect square. Go one step at a time; the trick is in the second step.

The proof does not depend on any particular x. So when 1 is added to the product of any four consecutive natural numbers, the result is a perfect square. If a, b, c, d are consecutive, abcd + 1 is a perfect square.

2. Even and odd, in letters

Even numbers are divisible by 2: 2, 4, 6, 8, ... are even natural numbers, with general form 2m. Odd numbers have the form 2m − 1 (m ∈ N), as in n = 2m − 1. And for n ∈ N, 2n + 1 is also odd.

The square of an even number is even. So if a, b are consecutive even numbers, a² and b² are even, b² + 2 is even, but a² + 1 is odd. The square of an odd number is odd: (2m − 1)² = 4m² − 4m + 1 = 2(2m² − 2m) + 1.

Take two consecutive even numbers 2m and 2m + 2. Their product is 2m(2m + 2) = 4m(m + 1). One of m and m + 1 is even, so m(m + 1) is even. The product is 4 times an even number, so it is divisible by 8.

Now sort for yourself. a, b are consecutive even numbers, and n is an odd natural number.

3. Divisibility by 8 and 6

If n = 2m − 1, then n² − 1 = (2m − 1)² − 1 = 4m² − 4m = 4m(m − 1). m(m − 1) is a product of two consecutive numbers, so it is even. Hence for odd n, n² − 1 is always divisible by 8.

Among three consecutive numbers at least one is even and one is a multiple of 3. So their product is always divisible by 6, for example 4 × 5 × 6 = 120.

4. Differences of squares, and making a square

a² + b² + 2ab = (a + b)², so adding 2ab to a² + b² gives a whole square.

Translating words into algebra is half the work. For example, adding a natural number x to its square gives x² + x.

For two consecutive numbers n and n + 1 the difference of squares is (n + 1)² − n² = 2n + 1. If it is 7 then 2n + 1 = 7, n = 3. The numbers are 3 and 4.

For two consecutive odd numbers n and n + 2 the difference of squares is (n + 2)² − n² = 4n + 4. If it is 72 then 4n + 4 = 72, n = 17, and the larger number is 19. Check: 19² − 17² = 361 − 289 = 72.

Now change the difference and watch: general expression first, then the equation. The number you type goes nowhere. The notebook is yours.

5. Check yourself

Three questions in the exam's style. Start with the first.

In the second, put the forms in letters to work.

In the last one, judge each statement in turn.

The whole note on seven cards.

Source: Mathematics, Classes 9-10 (NCTB), Chapter 1 “Real Numbers”, the “Proofs with Consecutive Natural Numbers” part.