6 min read
Inductive Step
Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
Choose Your Level
Inductive Step
What is it?
Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
In simple words
- Write P(k): the formula with n=k.
- Add the next term
- Or multiply the next factor
- State the rule, show the working, and check the answer.
Important terms
INDUCTIVE STEP: Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
STEP: Write P(k): the formula with n=k.
What you should learn
- State the rule or meaning of inductive step in easy words.
- Apply inductive step in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Examples
- Add the next term
- Or multiply the next factor
- End at the P(k+1) formula
MAIN IDEA
Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
EXAM USE
Write the rule first: Write P(k): the formula with n=k. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Recommended for Class 11
Full school-level explanation
Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
Understand the idea, then apply it:
WHAT — INDUCTIVE STEP
Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
WHY — Why it matters
This rule is a separate learning unit inside principle of mathematical induction. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Write P(k): the formula with n=k. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
MAIN IDEA
Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
Remember
- IH = P(k)
Examples
- Add the next term
- Or multiply the next factor
- End at the P(k+1) formula
EXAM USE
Write the rule first: Write P(k): the formula with n=k. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Remember
- focused answer
- accurate support
Examples
- State the central idea
- Use one accurate detail
- Check the final response
How to answer a inductive step question
How it works
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
Remember
- IH = P(k)
- base then step
- assume P(k), prove P(k+1)
- need the first domino
- IH = P(k)
- focused answer
- accurate support
- IH = P(k)
Deeper understanding
Go beyond memorising Inductive Step. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
What to Know
Example or Evidence
How to Answer
Exam focus
Quick revision
- IH = P(k)
- base then step
- assume P(k), prove P(k+1)
- need the first domino
- INDUCTIVE STEP: Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to
- STEP: Write P(k): the formula with n=k.
You must do both base and step. The step alone does not start the chain.
For long answers on Inductive Step: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- IH = P(k)
- base then step
- assume P(k), prove P(k+1)
- need the first domino
Important formulas
- IH = P(k)
- base then step
- assume P(k), prove P(k+1)
Common exam traps
- You must do both base and step. The step alone does not start the chain.
- Do not choose a formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
You must do both base and step. The step alone does not start the chain.
- Mistake
Do not choose a formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes, restrictions, or skipped algebra steps.
- Mistake
You must do both base and step. The step alone does not start the chain.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
IH = P(k)
base then step
assume P(k), prove P(k+1)
need the first domino
Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
IH = P(k)
- Definition
- Write P(k): the formula with n=k. For P(k+1), start from the left side, replace using P(k), tidy to the right side of P(k+1). Say 'by IH' when you use P(k).
- Formula
- IH = P(k)
- Example
- Add the next term
- Critical warning
- You must do both base and step. The step alone does not start the chain.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Ready to test yourself?
You've completed the learning section.
Exam Practice
Define
Q1. Define Inductive Step.
Explain
Q2. Explain: MAIN IDEA.
Example
Q3. Give one clear example of Inductive Step and say why it fits.
Mistake
Q4. State one common mistake students make about Inductive Step.
Exam note
Q5. Write a short exam note on Inductive Step: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.