Proofs

"Pure mathematics is, in its way, the poetry of logical ideas."

Albert Einstein

Recognize all the basic functions
You must be able to manipulate algebraic expressions.
(If not, check this lesson)
You must be able to manipulate exponential expressions.
(If not, check this lesson)
You must be able to manipulate logarithmic expressions.
(If not, check this lesson)

1. You should be able to use the principle of mathematical induction to prove statements about patterns involving integers. If you can show that the statement is true for \( n = 1 \), and that when the statement is true for \( n = k \) it is also true for \( n = k + 1 \), then the statement is true for all positive integers \( n \). The proof can be adapted for a starting value other than \( n = 1 \).
2. You should be able to use proof by contradiction: assume that the statement you are trying to prove is false and show that this leads to an impossible or contradictory conclusion.
3. You should be able to use counterexamples to disprove a statement. One counterexample is sufficient to prove that a statement is false.

IB Past Paper Problems

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