AQA · Mathematics · 8300
GCSE Maths: Algebraic Proof
Algebra · AQA GCSE Maths (8300). Cards are generated from the Corbettmaths textbook exercise, practice questions and answers for Algebraic Proof (Corbettmaths Video 365), and every card links back to the exact page it came from. Practice tests are drawn from this deck’s own cards. Source material © Corbettmaths — used with attribution.
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In algebraic proof, how do you represent a general even number?
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2n, where n is an integer.
In algebraic proof, how do you represent a general odd number?
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2n + 1 (or 2n − 1), where n is an integer.
How do you represent three consecutive integers algebraically?
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n, n + 1, n + 2.
Why is the sum of any three consecutive integers divisible by 3?
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n + (n+1) + (n+2) = 3n + 3 = 3(n + 1), which is 3 times an integer.
How do you represent two consecutive even numbers algebraically?
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2n and 2n + 2.
Prove the product of two odd numbers is always odd.
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(2n+1)(2m+1) = 4nm + 2n + 2m + 1 = 2(2nm + n + m) + 1, which is one more than an even number, so odd.
Why does the difference of the squares of two consecutive integers equal their sum?
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(n+1)² − n² = 2n + 1, which is n + (n + 1).
Squaring any odd integer gives a result that is one more than a multiple of ?.
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8
Why is (2n+1)² always one more than a multiple of 8?
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(2n+1)² = 4n² + 4n + 1 = 4n(n+1) + 1; n(n+1) is a product of consecutive integers so is even, making 4n(n+1) a multiple of 8.
What is the nth term of the sequence 2, 7, 12, 17, 22, …?
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5n − 3.
What is the nth term of the sequence 5, 11, 17, 23, 29, …?
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6n − 1.
How do you prove an expression like n² − 12n + 38 is always positive?
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Complete the square: (n − 6)² + 2. A square is ≥ 0, so the expression is ≥ 2 > 0.
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