Kenneth ChengMathematical modeling
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Prove 2ⁿ≥n+1 for every integer n≥t, using t as the base.

Read the idea, work independently, then explain what changed.

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TOPIC 01

Sequences: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

Prove 2ⁿ≥n+1 for every integer n≥t, using t as the base.

t=27t=27
  • Induction proves a statement only for the specified integer domain starting at the base case.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Verify the base case, assume the claim at k, and prove it at k+1.
Hint 2
Use this intermediate relation.
2(k+1)=2⋅2k2^(k+1)=2·2^k
Worked solution

  1. Verify the base case, assume the claim at k, and prove it at k+1.

  2. Apply the stated relation and retain its conditions.

    n=27:227=134217728≥28n=27:2^{27}=134217728≥28

  3. Apply the stated relation and retain its conditions.

    2k+1≥2(k+1)≥k+22^{k+1}≥2(k+1)≥k+2

  4. The explicit base check and the induction step cover only the stated domain.

The requested relation or conclusion is shown below.

2n≥n+1(n≥27)2^n\ge n+1\quad(n\ge27)

Checks and common pitfalls: The explicit base check and the induction step cover only the stated domain.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗