Prove algebraically that the sum of the squares of any two consecutive even numbers is always a multiple of 4. - Edexcel - GCSE Maths - Question 16 - 2019 - Paper 3
Question 16
Prove algebraically that the sum of the squares of any two consecutive even numbers is always a multiple of 4.
Worked Solution & Example Answer:Prove algebraically that the sum of the squares of any two consecutive even numbers is always a multiple of 4. - Edexcel - GCSE Maths - Question 16 - 2019 - Paper 3
Step 1
Let the two consecutive even numbers be $2n$ and $2n + 2$
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Answer
To prove the statement, we will start by expressing two consecutive even numbers using the variable n, where n is an integer. The first even number can be represented as 2n and the next consecutive even number is 2n+2.
Step 2
Calculate the sum of the squares of these two numbers
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Answer
The sum of the squares can be calculated as:
extSum=(2n)2+(2n+2)2
Calculating each square gives:
(2n)2=4n2(2n+2)2=4n2+8n+4
Thus, the sum becomes:
extSum=4n2+(4n2+8n+4)=8n2+8n+4
Step 3
Factor the expression
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Answer
Now we can factor out the common factor from the expression:
extSum=4(2n2+2n+1)
From this representation, it is evident that the expression is a multiple of 4, regardless of the integer value of n.
Step 4
Conclusion
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Answer
Therefore, we have shown that the sum of the squares of any two consecutive even numbers is always a multiple of 4.
