Question 1 - Page 115 - Sum of Consecutive Numbers - Chapter 5 Class 8 - Number Play (Ganita Prakash)

Transcript

Question 1 - Page 115 Is the phenomenon of all the expressions having the same parity limited to taking 4 numbers? What do you think?No. It works for any amount of numbers—whether you pick 2, 3, 5, or even 100 numbers! Here is the detailed explanation with examples to prove it. The "Why" (The Universal Rule) If you take any list of numbers (e.g., 1, 8, 9) and write out all the combinations of + and -, they will all be even or all be odd. They will never mix. Why? Because of the "Switching Trick" in Explanation 1. Changing a sign changes the answer by an even amount, so it can never switch from Odd to Even or vice versa. Example 1: Using just 3 Numbers Let's try it with numbers 1, 2, 3. Step 1: Find the maximum sum (all plus signs) 1+2+3=6 The result is Even. Step 2: Try changing signs According to the rule, since the first answer is Even, every other combination must also be Even. Let's test a few: 1+2−3=0 (Even) 1−2+3=2 (Even) 1−2−3=−4 (Even) Result: It worked! They are all Even. Example 2: Using 5 Numbers Let's try it with 1, 2, 3, 4, 5. Step 1: Find the maximum sum 1+2+3+4+5=15 The result is Odd. Step 2: Prediction Because the first sum is Odd, the rule says EVERY possible combination of pluses and minuses must result in an ODD number. Step 3: Test random combinations Let's flip some signs randomly: Try 1: 1+2−3+4+5=9 (Odd) Try 2: 1−2+3−4+5=3 (Odd) Try 3: 1−2−3−4−5=−13 (Odd) Result: It worked again! No matter what we did, the answer stayed Odd. The Final Conclusion You can think of it like a chain. If you have a chain of 4 numbers, and you swap a sign, the parity stays the same. If you add a 5th number to the chain, you are just adding one more number that follows the same "switch" rule. If you add a 6th number, it still follows the rule. So, YES, this works for any set of numbers, whether the list is short (2 numbers) or extremely long (1,000 numbers). General rule If the sum of all numbers is Even → All combinations will be Even. If the sum of all numbers is Odd → All combinations will be Odd.