Proof
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What does "show that" / "prove" require?
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All Flashcards in Topic 1.6
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1.6.111 cards
What does "show that" / "prove" require?
A chain of justified steps from what you know to the result. The marks are the reasons, not the final answer.
The golden rule of "show that" questions?
Start from the given (or one side) and work toward the target. Never start from the answer and work backwards.
Difference between = and ≡?
= (equation) is true for particular values you solve for; ≡ (identity) is true for ALL values.
How do you write an even and an odd integer in algebra?
Even = 2k, odd = 2k + 1, where k is an integer.
How do you represent consecutive integers?
n, n + 1, n + 2, … — start from n and add 1 each time.
Prove the sum of two odd numbers is even.
(2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1), which is even. Use different letters a, b.
How do you show a number is a multiple of k?
Manipulate it until you can take out a factor of k: write it as k × (an integer).
Why is n(n − 1) always even?
It is a product of two consecutive integers, and one of any two consecutive integers is even.
Why must you use different letters for two unknowns?
Reusing one letter (e.g. 2k + 1 twice) forces the two numbers to be equal, which breaks a general proof.
How should a proof end?
Reach the target exactly, then conclude in words — "… = 2m, which is even, so …". That sentence is often the last mark.
Sum of three consecutive integers is a multiple of what?
3: n + (n + 1) + (n + 2) = 3(n + 1).
1.6.29 cards
How do you write consecutive integers?
n, n + 1, n + 2 — they go up by 1. Use one starting letter.
Prove the sum of three consecutive integers is a multiple of 3.
n + (n+1) + (n+2) = 3n + 3 = 3(n + 1) = 3 × a whole number.
Why is the product of two consecutive integers even?
One of any two consecutive integers is even, and an even factor makes the product even.
How do you prove a number is a multiple of k?
Take out a factor of k: write it as k × (a whole number).
How do you prove something is NEVER a multiple of k?
Show it always leaves the same remainder: k × (whole) + r with r ≠ 0.
Are the squares of three consecutive integers a multiple of 3 when summed?
No — the sum is 3(n² + 2n + 1) + 2, so it always leaves remainder 2.
Sum of three consecutive integers = 3 × what?
3 × the middle integer (3n + 3 = 3(n + 1)).
Can you prove a 'for all n' statement by testing examples?
No — examples never prove 'for all'. Use algebra (n, n + 1, …) and reason generally.
First step in a consecutive-integer proof?
Name them with one letter (n, n + 1, n + 2), then add or multiply.
1.6.310 cards
Difference between = and ≡?
= (equation) is true for particular values you solve for; ≡ (identity) is true for ALL values, which you prove.
How do you prove an identity LHS ≡ RHS?
Transform ONE side (the busier one) step by step into the other. Never move terms across the ≡.
Why can't you prove an identity by substituting one value?
One value only checks that single case; an identity must hold for every x.
Which side should you start from?
The busier side — the one with brackets/powers to expand or fractions to combine.
Method for a polynomial identity?
Expand all brackets, then collect like terms until it matches the target.
What does (a − b)² expand to?
a² − 2ab + b² — don't forget the middle term −2ab.
Method for a rational (fraction) identity?
Put the side over a common denominator, combine the numerators, then simplify/factor.
Common denominator of 1/x and 1/(x + 1)?
x(x + 1) — the product of the two distinct denominators.
What does "hence" tell you in a later part?
Use the identity/result you just proved — don't re-derive it from scratch.
Prove (x + 3)² − (x − 3)² ≡ 12x.
Expand: (x² + 6x + 9) − (x² − 6x + 9) = 12x. The x² and 9 cancel.
Topic 1.6 study notes
Full notes & explanations for Proof
Math AA SL exam skills
Paper structures, command terms & tips
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