AS-Level Further Maths

Revision Session 1

• Roots of Polynomials

Revision Session 2

• Proof by induction

Revision Session 3

• Momentum and Impulse

Revision Session 4

• Review and Hooke’s Law

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From the specification and teaching guidance

Proof By Induction

- Divisibility

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- Sums of Series

- Proof by induction of the binomial theorem

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- Powers of matrices (covered in 2023, ‘Matrices and Sums of Series’)

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Divisibility Questions

Solution

1. Basis step: When n = 1, 9¹ – 1 = 8 which is divisible by 8, therefore true for n = 1

2. Assumption step: Assume true for n = k: 9^k – 1 is divisible by 8

3. Induction step: When n = k + 1:

4. Conclusion step: Hence true for n = k => true for k+1 and since true for n = 1, the result is proved by induction.

Divisibility Questions

From the specification and teaching guidance

Proof By Induction

- Divisibility

​

- Sums of Series

- Proof by induction of the binomial theorem

​

- Powers of matrices (covered in 2023, ‘Matrices and Sums of Series’)

​

Sums of series questions

2. Assumption step: Assume true for n = k:

3. Induction step: When n = k + 1:

4. Conclusion step: Hence true for n = k => true for k+1 and since true for n = 1, the result is proved by induction.

LHS = RHS therefore true for n = 1.

Sequence question

Solution

1. Basis step: When n = 1, x₁ = 2¹ + 1 = 3 which is correct so true for n = 1

2. Assumption step: Assume true for n = k: x_k = 2^k + k

3. Induction step: When n = k + 1:

4. Conclusion step: Hence true for n = k => true for k+1 and since true for n = 1, the result is proved by induction.

Proving the binomial theorem using proof by induction

Formulae Booklet

Proving the binomial theorem using proof by induction

Pascals Identity

Other things to remember:

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Proving the binomial theorem using proof by induction

Solution

1. Basis step: When n = 1, (a + b)¹ = a¹ + b¹ = a + b

so true for n = 1

3. Induction step: When n = k + 1:

4. Conclusion step: Hence true for n = k => true for k+1 and since true for n = 1, the result is proved by induction.

From the specification and teaching guidance

Proof By Induction

- Divisibility

​

- Sums of Series

- Proof by induction of the binomial theorem

​

- Powers of matrices (covered in 2023, ‘Matrices and Sums of Series’)

​

AS-Level Further Maths

Revision Session 1

• Roots of Polynomials

Revision Session 2

• Proof by induction

Revision Session 3

• Momentum and Impulse

Revision Session 4

• Review and Hooke’s Law

​