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

1. 1
   A student is proving by induction that $\sum_{r=1}^{n}r=\frac{n(n+1)}{2}$ for all positive integers $n$, and has already checked the case $n=1$.

   (a)

   In the inductive step, the student assumes that the result is true for $n=k$. What must the student then show?

   [1 mark]

   • AThat $\sum_{r=1}^{k}r=\frac{k(k+1)}{2}$
   • BThat the result is true for $n=k+2$
   • CThat the result is true for all positive integers $n$
   • DThat $\sum_{r=1}^{k+1}r=\frac{(k+1)(k+2)}{2}$

   (b)

   Which of these is a complete and correct conclusion to the proof?

   [1 mark]

   • AThe result is true for $n=1$, so it is true for all positive integers $n$.
   • BThe result is true for $n=1$, and if it is true for $n=k$ then it is true for $n=k+1$, so by mathematical induction it is true for all positive integers $n$.
   • CIf the result is true for $n=k$ then it is true for $n=k+1$, so it is true for all positive integers $n$.
   • DThe result is true for $n=k+1$, so it is true for all positive integers $n$.

   (c)

   Assuming the result is true for $n=k$, show that it is true for $n=k+1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(n)=7^n-1$ for positive integers $n$.

   (a)

   Find $f(2)$.

   [1 mark]

   • A$48$
   • B$13$
   • C$50$
   • D$36$

   (b)

   Find $f(k+1)-f(k)$.

   [1 mark]

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

   (c)

   Hence, or otherwise, show that if $f(k)$ is divisible by $6$ then $f(k+1)$ is divisible by $6$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence is defined by $u_1=2$ and $u_{n+1}=2u_n+3$ for $n\ge1$.

   (a)

   Find $u_2$, $u_3$ and $u_4$, and verify that the formula $u_n=5\times2^{n-1}-3$ gives the correct values for $n=1,2,3,4$.

   [3 marks]

   (b)

   Prove by induction that $u_n=5\times2^{n-1}-3$ for all positive integers $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathbf{M}=\begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix}$.

   (a)

   Prove by induction that $\mathbf{M}^n=\begin{pmatrix} 2n+1 & -4n \\ n & 1-2n \end{pmatrix}$ for all positive integers $n$.

   [6 marks]

   (b)

   Use the result of part (a) to:
   (i) write down $\mathbf{M}^{12}$;
   (ii) show that $\det\left(\mathbf{M}^n\right)=1$ for all positive integers $n$;
   (iii) hence write down $\left(\mathbf{M}^n\right)^{-1}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions