Practise

Question bank

Original exam-style questions, mapped to your syllabus. Answers are checked for mathematical equivalence and every attempt updates your mastery.

TopicSkillDifficulty

12 questions

SA-FP-IND-0021 markL1 FoundationNon-calculatorDivisibility

To prove

3^{2n}+11

is divisible by

4

, first check

n=1

. Find

3^{2}+11

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SA-FP-IND-0052 marksL1 FoundationCalculatorSeries

Given

\displaystyle\sum_{r=1}^n r^3=\tfrac14n^2(n+1)^2

, evaluate

\displaystyle\sum_{r=1}^{10} r^3

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SA-FP-IND-Q011 markL1 FoundationNon-calculatorSeries

In an induction proof, which step uses the assumption?

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SA-FP-IND-X011 markL1 FoundationNon-calculatorDivisibility

Find the largest integer that divides

7^n-1

for every positive integer

n

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SA-FP-IND-X021 markL1 FoundationNon-calculatorSeries

It can be proved by induction that

\sum_{r=1}^nr(r+1)=\frac13n(n+1)(n+2)

. Use this to find

\sum_{r=1}^{10}r(r+1)

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SA-FP-IND-X041 markL1 FoundationNon-calculatorSeries

In the inductive step of a proof by induction you

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SA-FP-IND-0012 marksL2 StandardNon-calculatorDivisibility

Let

f(n)=7^n-1

. Find the constant

c

such that

f(k+1)-7f(k)=c

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SA-FP-IND-0032 marksL2 StandardNon-calculatorMatrices and sequences

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

and it can be proved that

\mathbf M^n=\begin{pmatrix}1&2n\\0&1\end{pmatrix}

. Find the top-right entry of

\mathbf M^5

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SA-FP-IND-0042 marksL2 StandardNon-calculatorMatrices and sequences

A sequence has

u_1=2

and

u_{n+1}=3u_n-2

. It can be proved that

u_n=3^{n-1}+1

. Find

u_4

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SA-FP-IND-G012 marksL2 StandardNon-calculatorSeries

In proving

\displaystyle\sum_{r=1}^n r(r+1)=\tfrac13n(n+1)(n+2)

, the sum to

k+1

terms is

\tfrac13k(k+1)(k+2)+(k+1)(k+2)

. This factorises to

\tfrac13(k+1)(k+2)(k+a)

. Find

a

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SA-FP-IND-X032 marksL2 StandardNon-calculatorMatrices and sequences

u_1=1

and

u_{n+1}=3u_n+2

. It can be shown by induction that

u_n=2\times3^{n-1}-1

. Find

u_5

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SA-FP-IND-0062 marksL4 ChallengingNon-calculatorMatrices and sequences

The statement

2^n>n^2

is to be proved by induction for all integers

n\ge n_0

. Find the smallest

n_0\ge3

for which the basis case is true.

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