Prove that the difference between two consecutive square numbers is always odd. - OCR - GCSE Maths - Question 18 - 2017 - Paper 1
Question 18
Prove that the difference between two consecutive square numbers is always odd.
Worked Solution & Example Answer:Prove that the difference between two consecutive square numbers is always odd. - OCR - GCSE Maths - Question 18 - 2017 - Paper 1
Step 1
Step 1: Define Consecutive Square Numbers
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Answer
Let the two consecutive integers be n and n+1. The square of n is given by n2, and the square of n+1 is given by (n+1)2.
Step 2
Step 2: Calculate the Difference
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Answer
The difference between the square of (n+1) and the square of n is calculated as follows:
(n+1)2−n2=(n2+2n+1)−n2=2n+1.
Step 3
Step 3: Analyze the Result
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Answer
The expression 2n+1 represents an odd number since it can be expressed as the sum of an even number (2n) and 1.
Step 4
Step 4: Conclusion
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Answer
Therefore, the difference between two consecutive square numbers is always odd, as derived from the expression 2n+1.
