Ayeesha took three sets of consecutive numbers \(7, 8, 9, \& 10\), \(9, 10, 11, \& 12\) and \(11, 12, 13 \& 14\). Place \(+\) and \(-\) signs between them, and then the results are as follows:
| Expression \(1\) | Expression \(2\) | Expression \(3\) |
| \(7 + 8 + 9 + 10 = 34\) | \(9 + 10 + 11 + 12 = 42\) | \(11 + 12 + 13 + 14 = 50\) |
| \(7 + 8 + 9 - 10 = 14\) | \(9 + 10 + 11 - 12 = 18\) | \(11 + 12 + 13 - 14 = 22\) |
| \(7 + 8 - 9 + 10 = 16\) | \(9 + 10 - 11 + 12 = 20\) | \(11 + 12 - 13 + 14 = 24\) |
| \(7 - 8 + 9 + 10 = 18\) | \(9 - 10 + 11 + 12 = 22\) | \(11 - 12 + 13 + 14 = 26\) |
| \(7 + 8 - 9 - 10 = -4\) | \(9 + 10 - 11 - 12 = -4\) | \(11 + 12 - 13 - 14 = -4\) |
| \(7 - 8 - 9 + 10 = 0\) | \(9 - 10 - 11 + 12 = 0\) | \(11 - 12 - 13 + 14 = 0\) |
| \(7 - 8 + 9 - 10 = -2\) | \(9 - 10 + 11 - 12 = -2\) | \(11 - 12 + 13 - 14 = -2\) |
| \(7 - 8 - 9 - 10 = -20\) | \(9 - 10 - 11 - 12 = -24\) | \(11 - 12 - 13 - 14 = -28\) |
1. Do these patterns occur no matter which \(4\) consecutive numbers are chosen? Is there a way to find out through reasoning? (Hint: Use algebra and describe the \(8\) expressions in a general form.)
2. Is there a way to explain why this happens? (Hint: Think of the rules for parity of the sum or difference of two numbers.)
3. Replace any negative sign in the expression \(a + b – c – d\) with a positive sign and subtract the modified expression from the orginal expression.
4. What do you conclude from this observation?