Proof questions

Questions to attempt

3 questions on Proof, each with its mark scheme and worked method. Reveal the answer when you have had a go.

1. CP-1.1 · Construct proofs using mathematical induction. Contexts include sums of series, divisibility and powers of matrices.

   Q1

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   1.

   Prove by mathematical induction that $8^n-1$ is divisible by $7$ for every positive integer $n$.

   (5)

   (Total for Question 1 is 5 marks)

   Reveal answer and method

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2. CP-1.1 · Construct proofs using mathematical induction. Contexts include sums of series, divisibility and powers of matrices.

   Q2

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   2.

   Prove by mathematical induction that $1+3+5+\cdots+(2n-1)=n^2$ for every positive integer $n$.

   (4)

   (Total for Question 2 is 4 marks)

   Reveal answer and method

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3. CP-1.1 · Construct proofs using mathematical induction. Contexts include sums of series, divisibility and powers of matrices.

   Q3

   ORIGINAL

   Year 1

   Year 1: this is the AS subject content the exam board publishes, which is what most schools teach in Year 12.

   3.

   Let $A=\begin{pmatrix}2&1\\0&2\end{pmatrix}$. Prove by mathematical induction that $A^n=\begin{pmatrix}2^n&n2^{n-1}\\0&2^n\end{pmatrix}$ for every positive integer $n$.

   (6)

   (Total for Question 3 is 6 marks)

   Reveal answer and method

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CP-1 Proof revision notes