1. CBQ - Consecutive numbers and even numbers

Ayeesha took three sets of consecutive numbers \(4, 5, 6, \& 7\), \(6, 7, 8, \& 9\) and \(8, 9, 10 \& 11\). Place \(+\) and \(-\) signs between them, and then the results are as follows:

 

Expression \(1\)	Expression \(2\)	Expression \(3\)
\(4 + 5 + 6 + 7 = 22\)	\(6 + 7 + 8 + 9 = 30\)	\(8 + 9 + 10 + 11 = 38\)
\(4 + 5 + 6 - 7 = 8\)	\(6 + 7 + 8 - 9 = 12\)	\(8 + 9 + 10 - 11 = 16\)
\(4 + 5 - 6 + 7 = 10\)	\(6 + 7 - 8 + 9 = 14\)	\(8 + 9 - 10 + 11 = 18\)
\(4 - 5 + 6 + 7 = 12\)	\(6 - 7 + 8 + 9 = 16\)	\(8 - 9 + 10 + 11 = 20\)
\(4 + 5 - 6 - 7 = -4\)	\(6 + 7 - 8 - 9 = -4\)	\(8 + 9 - 10 - 11 = -4\)
\(4 - 5 - 6 + 7 = 0\)	\(6 - 7 - 8 + 9 = 0\)	\(8 - 9 - 10 + 11 = 0\)
\(4 - 5 + 6 - 7 = -2\)	\(6 - 7 + 8 - 9 = -2\)	\(8 - 9 + 10 - 11 = -2\)
\(4 - 5 - 6 - 7 = -14\)	\(6 - 7 - 8 - 9 = -18\)	\(8 - 9 - 10 - 11 = -22\)

 

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 Obtain the difference between the two numbers.

 

4. What do you conclude from this observation?