Consecutive numbers:
Consecutive numbers are numbers that follow each other in order without any gaps, where each number is exactly \(1\) more than the previous number. For example: \(5, 6, 7, 8\) are consecutive numbers.
General form of consecutive numbers:
Any consecutive number can be written as \(x\), (\(x + 1\)), (\(x + 2\)), (\(x + 3\)), ....
Any even consecutive number can be written as \(2x\), \(2x + 2\), \(2x + 4\), ....
Any odd consecutive number can be written as \(2x+1\), \(2x + 3\), \(2x + 5\), ....
Magic of \(+\) and \(-\) with \(4\) consecutive numbers:
Let's take four consecutive numbers as \(3\), \(4\), \(5\) and \(6\).
Now, we place \(+\) or \(-\) signs between them.
- \(3 + 4 + 5 + 6 = 18\)
- \(3 + 4 + 5 - 6 = 6\)
- \(3 + 4 - 5 + 6 = 8\)
- \(3 - 4 + 5 + 6 = 10\)
- \(3 - 4 - 5 + 6 = 0\)
- \(3 + 4 - 5 - 6 = -4\)
- \(3 - 4 + 5 - 6 = -2\)
- \(3 - 4 - 5 - 6 = -12\)
Here, note that sum of all the possibilities of consecutive numbers are even number.
Thus, sum of any four consecutive numbers is even.
Important!
Every odd number can be written as the sum of two consecutive numbers.
Ex: \(5 = 2 + 3\)
Every numbers cannot always be written as sum of two consecutive numbers.
Ex: \(8 = 1 + 2 + 5\) or \(8 = 1 + 3 + 4\), which are not consecutive.