Proof and Reasoning | IB Math AAHL | Perplex

Proof and Reasoning

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Watch comprehensive video reviews for Proof and Reasoning, designed for final exam preparation. Each video includes integrated problems you can solve alongside detailed solutions.

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Not your average video:

Interactive Problems: Solve problems alongside the video with step-by-step guidance and detailed solutions.

Exam Preparation: Complete unit reviews designed for final exam preparation with all key concepts covered systematically.

Expert Teaching: High-quality instruction from Perplex co-founder James Mullen with clear explanations, worked examples, and exam tips.

Direct proof (SL + HL)HL Terms & DefinitionsContradiction (HL)Proof by induction

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Induction with summation

AHL AA 1.15

Prove that

r=1∑nr=2n(n+1)

for all n∈Z+.

Let P(n) be the proposition that

r=1∑nr=2n(n+1)

Verify P(1):

For n=1, the left-hand side is 1 and the right-hand side is 21⋅2=1. Hence, P(1) is true.

Assume P(k) is true:

r=1∑kr=2k(k+1)

Show that P(k+1) is true:

r=1∑k+1r=r=1∑kr+(k+1)=2k(k+1)+(k+1)

Factor out (k+1):

2k(k+1)+(k+1)=(k+1)(2k+1)=2(k+1)(k+2).

Thus, P(k+1) holds.

Conclusion:

Since P(1) is true and if P(k) is true then P(k+1) is true, P(n) is true for all n∈Z+.

Induction with summation

AHL AA 1.15

Prove that

r=1∑nr=2n(n+1)

for all n∈Z+.

Let P(n) be the proposition that

r=1∑nr=2n(n+1)

Verify P(1):

For n=1, the left-hand side is 1 and the right-hand side is 21⋅2=1. Hence, P(1) is true.

Assume P(k) is true:

r=1∑kr=2k(k+1)

Show that P(k+1) is true:

r=1∑k+1r=r=1∑kr+(k+1)=2k(k+1)+(k+1)

Factor out (k+1):

2k(k+1)+(k+1)=(k+1)(2k+1)=2(k+1)(k+2).

Thus, P(k+1) holds.

Conclusion:

Since P(1) is true and if P(k) is true then P(k+1) is true, P(n) is true for all n∈Z+.