Further Mathematics · OCR H245 · Topic 4.01

Proof

Mathematical induction.

1 sub-topic
Taught in Year 12
14 practice questions
1 animated lesson

Practice questions (14) Lessons Worksheets coming soon All OCR Further Mathematics topics

What the specification says

Every sub-topic OCR can examine in this topic, with its reference number from the official specification. The Y12 and Y13 tags show when each sub-topic is usually taught.

4.01a-bMathematical inductionY12

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Easy, medium and hard questions from this topic, each with a worked solutions and the mark for every step. All 14 questions in this topic, free, no account needed.

Proof

Easy

Induction: matrix powers · Year 12 · 4 marks

\mathbf{M}

{}= \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}

Prove by induction that $\mathbf{M}^n$ ${}= \begin{pmatrix} 1 & 2n \\ 0 & 1 \end{pmatrix}$ .
(4)

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$n = 1$ gives $\mathbf{M}$ .
The inductive hypothesis: no mark on its own; the mark is for using it.Assume true for $n = k$ .
Multiply. $\mathbf{M}^{k+1}$ ${}= \begin{pmatrix} 1 & 2k \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ ${}= \begin{pmatrix} 1 & 2(k + 1) \\ 0 & 1 \end{pmatrix}$
So true for all positive integers $n$ by induction.

Total4

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Proof

Medium

Induction: a series result · Year 12 · 5 marks

Prove by induction that $\sum_{r=1}^{n} r^3$ ${}= \tfrac{1}{4}n^2(n + 1)^2$ for all positive integers $n$ .
(5)

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Basis: check $n = 1$ . $n = 1: \ 1 = \tfrac{1}{4}(1)(4)$
The inductive hypothesis: no mark on its own; the mark is for using it.Assume true for $n = k$ .
Add the $(k + 1)$ th term to the assumed sum and factorise. $\tfrac{1}{4}k^2(k + 1)^2$ ${}+ (k + 1)^3$ ${}= \tfrac{1}{4}(k + 1)^2(k^2 + 4k + 4)$
The conclusion must be stated in full. $= \tfrac{1}{4}(k + 1)^2(k + 2)^2$ . True for $n = 1$ and true for $n = k$ ⇒ true for $n = k + 1$ , so true for all $n$ by induction.

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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Proof

Hard

Induction: divisibility · Year 12 · 5 marks

Prove by induction that $5^n + 3$ is divisible by 4 for all positive integers $n$ .
(5)

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Basis. $n = 1: \ 8 = 4 \times 2$
The inductive hypothesis: no mark on its own; the mark is for using it.Let $f(n) = 5^n + 3$ and assume $f(k) = 4m$ for an integer $m$ .
Express $f(k + 1)$ using $f(k)$ . $f(k + 1)$ ${}= 5^{k+1} + 3$ ${}= 5(5^k + 3) - 12$ ${}= 20m - 12$
The conclusion must be stated in full. $= 4(5m - 3)$ , a multiple of 4. True for $n = 1$ and true for $n = k$ ⇒ true for $n = k + 1$ , so true for all $n$ by induction.

Total5

MMethod mark: for a correct method, even with a slip.
AAccuracy mark: for a correct answer from a correct method.
BIndependent mark: for a correct fact or result on its own, with no method needed.

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What the specification says

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