Exam Response (induction and Binomial)

Exam Response (induction and Binomial)

What is it?

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.

In simple words

• 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.

Important terms

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).

What you should learn

• 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.

Examples

• P(1)
• IH
• P(k+1)

INDUCTION

Write P(n) clearly. Base number check. Assume P(k). Add/multiply the next piece. Conclude for all n>=n0.

BINOMIAL

General term. Solve r. n even/odd for middle. (1+x)^n coeff nCk. Signs from (a-b).

Full school-level explanation

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.

Proof, then expand:

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

INDUCTION

Write P(n) clearly. Base number check. Assume P(k). Add/multiply the next piece. Conclude for all n>=n0.

Remember

• three labels

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).

Remember

• r from exponent

Examples

• T_{r+1}
• nCr
• (-1)^r

How to answer a exam response (induction and binomial) question

How it works

• 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.

Remember

• base + IH + P(k+1)
• powers add to n
• 160
• (2x)^r = 2^r x^r
• three labels
• r from exponent
• base + IH + P(k+1)
• powers add to n

How the ideas connect

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

Deeper understanding

Go beyond memorising Exam Response (induction and Binomial). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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

Common mistakes

Exam focus

Core Idea

1. INDUCTION

2. BINOMIAL

Worked Example

Exam focus

Quick revision

• base + IH + P(k+1)
• powers add to n
• 160
• (2x)^r = 2^r x^r
• EXAM RESPONSE (INDUCTION AND BINOMIAL): Typical tasks: full induction (base+step); T_{r+1}; middle term; coefficient of x^k; two-term approx
• 1. INDUCTION: Write P(n) clearly. Base number check. Assume P(k). Add/multiply the next piece. Conclude for all n>

(2x)^3 is 8 x^3, not 2 x^3. Raise the coefficient too.

For long answers on Exam Response (induction and Binomial): define clearly → explain the main parts → give one example → state one exam caution.