Jovan has made the following discovery for three consecutive natural numbers:
If you multiply the first number by the third, the result is 1 less than the square of the middle number.
Example:
4, 5, 6: 4 × 6 = 52 – 1 = 24
7, 8, 9: 7 × 9 = 82 – 1 = 63
10, 11, 12: 10 × 12 = 112 – 1 = 120
Does this rule apply to all natural numbers?
Yes, the rule applies to all natural numbers.
For the natural numbers a, a + 1 and a + 2, the following applies:
a × (a + 2) = (a + 1)2 – 1
The terms on both sides of the equation are equivalent:
a2 + 2a = a2 + 2a + 1 – 1
a2 + 2a = a2 + 2a
No matter which natural number you use in the equation, the rule always applies.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission.