Proof by mathematical inductionEdexcel A-Level Further Maths: Subtopic test

1. 1
   Proof by mathematical induction is used to show that $\sum_{r=1}^{n}\frac{1}{r(r+1)}=\frac{n}{n+1}$ for all positive integers $n$.

   (a)

   What is the value of each side of the identity when $n=1$?

   [1 mark]

   • A$1$
   • B$\frac12$
   • C$\frac13$
   • D$2$

   (b)

   Assuming the result is true for $n=k$, which expression is equal to the sum of the first $k+1$ terms?

   [1 mark]

   • A$\frac{k}{k+1}+\frac{1}{k(k+1)}$
   • B$\frac{k+1}{k+2}+\frac{1}{(k+1)(k+2)}$
   • C$\frac{k}{k+1}+\frac{1}{(k+1)(k+2)}$
   • D$\frac{k}{k+1}+\frac{1}{k+2}$

   (c)

   Show that if the result is true for $n=k$ then it is true for $n=k+1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\mathrm{f}(n)=7^n+5$, where $n$ is a positive integer.

   (a)

   Find $\mathrm{f}(2)$.

   [1 mark]

   • A$54$
   • B$19$
   • C$24$
   • D$49$

   (b)

   Which expression is equal to $\mathrm{f}(k+1)-\mathrm{f}(k)$?

   [1 mark]

   • A$7^k$
   • B$8\times7^k$
   • C$7^{k+1}+5$
   • D$6\times7^k$

   (c)

   Given that $\mathrm{f}(k)$ is divisible by 6 for some positive integer $k$, show that $\mathrm{f}(k+1)$ is also divisible by 6.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{A}=\begin{pmatrix}2&1\\0&1\end{pmatrix}$.

   (a)

   Find $\mathbf{A}^2$ and $\mathbf{A}^3$, and hence suggest a formula for $\mathbf{A}^n$.

   [3 marks]

   (b)

   Prove by induction that your formula for $\mathbf{A}^n$ is correct for all positive integers $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question you must use proof by mathematical induction, stating the assumption made at the inductive step and ending with a clear conclusion. $n$ is a positive integer.

   (a)

   Prove that $\sum_{r=1}^{n}r(r+2)=\frac{n(n+1)(2n+7)}{6}$.

   [6 marks]

   (b)

   Prove that $2^{n+2}+3^{2n+1}$ is divisible by 7.

   [6 marks]

   Total for question 4: 12 marks

End of questions