MathematicsChapter 7: Mathematical Induction and Binomial Theorem
Exam Response (induction and Binomial)
Typical tasks: full induction (base+step); T_{r+1}; middle term; coefficient of x^k; two-term approximation.
Learning objectives
- State the main ideas and notation used in exam response (induction and binomial).
- Apply the rules of exam response (induction and binomial) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Typical tasks: full induction (base+step); T_{r+1}; middle term; coefficient of x^k; two-term approximation. For induction, label Base, IH, Aim, then algebra. For binomial, find r from the exponent.
Mathematical form
b=2x, r=3
b=2x
r=3 6C3 (1)3 (2x)3
Where / symbols
- EXAM RESPONSE (INDUCTION AND BINOMIAL)
- Typical tasks: full induction (base+step); T_{r+1}; middle term; coefficient of x^k; two-term approximation. For induction, label Base, IH, Aim, then algebra. For binomial, find r from the exponent.
- 1. INDUCTION
- Write P(n) clearly. Base number check. Assume P(k). Add/multiply the next piece. Conclude for all n>=n0.
- 2. BINOMIAL
- General term. Solve r. n even/odd for middle. (1+x)^n coeff nCk. Signs from (a-b).
Overview
Proof, then expand:
Induction
Write P(n) clearly. Base number check. Assume P(k). Add/multiply the next piece. Conclude for all n>=n0.
Examples P(1) · IH · P(k+1)
Binomial
General term. Solve r. n even/odd for middle. (1+x)^n coeff nCk. Signs from (a-b).
Examples T_{r+1} · nCr · (-1)^r
WHAT — EXAM RESPONSE (INDUCTION AND BINOMIAL)
Typical tasks: full induction (base+step); T_{r+1}; middle term; coefficient of x^k; two-term approximation. For induction, label Base, IH, Aim, then algebra. For binomial, find r from the exponent.
RULE — 1. INDUCTION
Write P(n) clearly. Base number check. Assume P(k). Add/multiply the next piece. Conclude for all n>=n0.
METHOD — 2. BINOMIAL
General term. Solve r. n even/odd for middle. (1+x)^n coeff nCk. Signs from (a-b).
WORKED EXAMPLE
Worked 4-mark coeff of x^3 in (1+2x)^6. b=2x, r=3 6C3 (1)^3 (2x)^3 = 20 * 8 x^3 = 160 x^3 coefficient 160
How to answer a exam response (induction and binomial) question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- Worked 4-mark coeff of x^3 in (1+2x)^6.
- b=2x, r=3
- 6C3 (1)^3 (2x)^3
- = 20 * 8 x3 = 160 x^3
- coefficient 160
In simple words
(2x)^3 is 8 x^3, not 2 x^3. Raise the coefficient too.
Key points
- base + IH + P(k+1)
- powers add to n
- three labels
- r from exponent
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Exam Response (induction and Binomial)
Sticky Notes · Quick Revision
Core Idea
- Typical tasks: full induction (base+step); T_{r+1}; middle term; coefficient of x^k; two-term approximation.
- For induction, label Base, IH, Aim, then algebra.
- For binomial, find r from the exponent.
1. INDUCTION
- Write P(n) clearly.
- Base number check.
- Assume P(k).
- Add/multiply the next piece.
2. BINOMIAL
- General term.
- Solve r.
- n even/odd for middle.
- (1+x)^n coeff nCk.
Worked Example
- Worked 4-mark coeff of x^3 in (1+2x)^6.
- b=2x, r=3
- 6C3 (1)^3 (2x)^3
- = 20 * 8 x3 = 160 x^3
- coefficient 160
Common Misconception
- (2x)^3 is 8 x^3, not 2 x^3. Raise the coefficient too.
FBISE Exam Check
- State the rule for exam response (induction and binomial) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Typical tasks: full induction (base+step); T_{r+1}; middle term; coefficient of x^k; two-term approximation.
- For induction, label Base, IH, Aim, then algebra.
- For binomial, find r from the exponent.
- (2x)^3 is 8 x^3, not 2 x^3. Raise the coefficient too.
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