Core Pure: ProofEdexcel A-Level Further Maths: Topic test

1. 1
   Proof by mathematical induction is to be used to show that $\sum_{r=1}^{n}r^3=\frac14n^2(n+1)^2$ for all positive integers $n$.

   (a)

   What is the value of the left-hand side when $n=3$?

   [1 mark]

   • A$27$
   • B$36$
   • C$216$
   • D$14$

   (b)

   Assuming the result is true for $n=k$, which expression is equal to $\sum_{r=1}^{k+1}r^3$?

   [1 mark]

   • A$\frac14k^2(k+1)^2+k^3$
   • B$\frac14k^2(k+1)^2+(k+1)$
   • C$\frac14k^2(k+1)^2+(k+1)^3$
   • D$\frac14(k+1)^2(k+2)^2+(k+1)^3$

   (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
   The function $\mathrm{g}(n)=8^n+13$ is defined for positive integers $n$.

   (a)

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

   [1 mark]

   • A$77$
   • B$29$
   • C$21$
   • D$64$

   (b)

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

   [1 mark]

   • A$8^k$
   • B$9\times8^k$
   • C$8^{k+1}+13$
   • D$7\times8^k$

   (c)

   Given that $\mathrm{g}(k)$ is divisible by 7 for some positive integer $k$, show that $\mathrm{g}(k+1)$ is also divisible by 7.

   [2 marks]

   Total for question 2: 4 marks

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

   (a)

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

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Throughout this question, $n$ is a positive integer and every proof must state the inductive assumption and end with a conclusion.

   (a)

   Prove by induction that $\sum_{r=1}^{n}r\times r!=(n+1)!-1$.

   [6 marks]

   (b)

   The matrix $\mathbf{R}=\begin{pmatrix}2&3\\0&2\end{pmatrix}$. Prove by induction that $\mathbf{R}^n=\begin{pmatrix}2^n&3n\times2^{n-1}\\0&2^n\end{pmatrix}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Proof by mathematical induction is to be used to show that $\sum_{r=1}^{n}r(r+1)(r+2)=\frac14n(n+1)(n+2)(n+3)$ for all positive integers $n$.

   (a)

   What is the value of the right-hand side of the formula when $n=2$?

   [1 mark]

   • A$24$
   • B$6$
   • C$120$
   • D$30$

   (b)

   Which expression is the right-hand side of the formula with $n=k+1$?

   [1 mark]

   • A$\frac14(k+1)(k+2)(k+3)(k+4)$
   • B$\frac14k(k+1)(k+2)(k+3)+1$
   • C$\frac14(k+1)(k+2)(k+3)$
   • D$\frac14k(k+2)(k+3)(k+4)$

   (c)

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

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $\mathrm{h}(n)=7^n-3^n$, where $n$ is a positive integer.

   (a)

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

   [1 mark]

   • A$8$
   • B$40$
   • C$4$
   • D$58$

   (b)

   Which expression is equal to $\mathrm{h}(k+1)-7\mathrm{h}(k)$?

   [1 mark]

   • A$-10\times3^k$
   • B$-4\times3^k$
   • C$4\times3^k$
   • D$4\times7^k$

   (c)

   Given that $\mathrm{h}(k)$ is divisible by 4 for some positive integer $k$, show that $\mathrm{h}(k+1)$ is also divisible by 4.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A sequence is defined by $u_1=1$ and $u_{n+1}=2u_n+3$ for $n\geqslant1$.

   (a)

   Find $u_2$, $u_3$ and $u_4$, and show that each agrees with the formula $u_n=2^{n+1}-3$.

   [3 marks]

   (b)

   Prove by induction that $u_n=2^{n+1}-3$ for all positive integers $n$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Throughout this question, $n$ is a positive integer and each result must be proved by mathematical induction.

   (a)

   Prove that $\mathrm{f}(n)=2^{n+2}+3^{2n+1}$ is divisible by 7.

   [6 marks]

   (b)

   Prove that $\sum_{r=1}^{n}\frac{r}{2^r}=2-\frac{n+2}{2^n}$.

   [6 marks]

   Total for question 8: 12 marks

End of questions