FP1: ProofEdexcel International A Level Further Maths: Topic test

1. 1
   Proof by induction is used to show that $\sum_{r=1}^{n}r^2=\frac{n(n+1)(2n+1)}{6}$ for all positive integers $n$.

   (a)

   The result is assumed true for $n=k$. Which expression is the sum of the first $k+1$ square numbers, written using this assumption?

   [1 mark]

   • A$\frac{k(k+1)(2k+1)}{6}+k^2$
   • B$\frac{k(k+1)(2k+1)}{6}+(k+1)^2$
   • C$\frac{k(k+1)(2k+1)}{6}+(k+1)$
   • D$\frac{(k+1)(k+2)(2k+3)}{6}+(k+1)^2$

   (b)

   In the inductive step the expression $\frac{k(k+1)(2k+1)}{6}+(k+1)^2$ is written as $\frac{k+1}{6}\left[\;\cdots\;\right]$. What is the expression in the bracket?

   [1 mark]

   • A$2k^2+k+6$
   • B$2k^2+7k+1$
   • C$2k^2+13k+12$
   • D$2k^2+7k+6$

   (c)

   Show that the result is true for $n=1$, and write down the conclusion that completes the proof once the inductive step has been shown.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $f(n)=5^n+3$ for positive integers $n$. It is to be proved by induction that $f(n)$ is divisible by $4$.

   (a)

   Which expression is equal to $f(k+1)-f(k)$?

   [1 mark]

   • A$4\times5^k$
   • B$5^k$
   • C$4\times5^{k+1}$
   • D$5^{k+1}+3$

   (b)

   Which statement is the correct inductive hypothesis?

   [1 mark]

   • A$f(n)$ is divisible by $4$ for all positive integers $n$.
   • B$f(k+1)$ is divisible by $4$.
   • C$f(k)$ is divisible by $4$ for some positive integer $k$.
   • D$f(1)$ is divisible by $4$.

   (c)

   Show that $f(k+1)=5f(k)-12$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A sequence is defined by $u_1=5$ and $u_{n+1}=3u_n-4$ for $n\ge1$.

   (a)

   Find $u_2$ and $u_3$, and verify that the formula $u_n=3^n+2$ gives the correct values of $u_1$, $u_2$ and $u_3$.

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $\mathbf{B}=\begin{pmatrix}2&1\\ 0&1\end{pmatrix}$.

   (a)

   Prove by induction that $\mathbf{B}^n=\begin{pmatrix}2^n&2^n-1\\ 0&1\end{pmatrix}$ for all positive integers $n$.

   [6 marks]

   (b)

   Use the result in part (a) to answer the following. (i) Write down $\mathbf{B}^{10}$. (ii) Find the least positive integer $n$ for which the top-right entry of $\mathbf{B}^n$ is greater than $10^6$. (iii) Find $\det\mathbf{B}^n$ in terms of $n$ and state what it represents for the transformation $\mathbf{B}^n$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $P(n)$ be the statement $\sum_{r=1}^{n}r(r+1)(r+2)=\frac{n(n+1)(n+2)(n+3)}{4}$.

   (a)

   What are the values of the left-hand side and the right-hand side of $P(1)$?

   [1 mark]

   • ALHS $=6$ and RHS $=24$
   • BBoth equal $24$
   • CLHS $=1$ and RHS $=6$
   • DBoth equal $6$

   (b)

   Assuming $P(k)$, which expression must be added to $\frac{k(k+1)(k+2)(k+3)}{4}$ to give the sum of the first $k+1$ terms?

   [1 mark]

   • A$k(k+1)(k+2)$
   • B$(k+1)(k+2)(k+3)$
   • C$(k+2)(k+3)(k+4)$
   • D$\frac{(k+1)(k+2)(k+3)(k+4)}{4}$

   (c)

   Show that $\frac{k(k+1)(k+2)(k+3)}{4}+(k+1)(k+2)(k+3)=\frac{(k+1)(k+2)(k+3)(k+4)}{4}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Let $h(n)=3^{2n}+7$ for positive integers $n$. It is to be proved by induction that $h(n)$ is divisible by $8$.

   (a)

   Find $h(2)$.

   [1 mark]

   • A$16$
   • B$81$
   • C$88$
   • D$43$

   (b)

   In the inductive step it is shown that $h(k+1)=9h(k)-56$. Which statement correctly completes the step?

   [1 mark]

   • ASince $h(k)$ is divisible by $8$ by assumption and $56=8\times7$, both terms are multiples of $8$, so $h(k+1)$ is divisible by $8$.
   • BSince $56=8\times7$ is a multiple of $8$, $h(k+1)$ is divisible by $8$.
   • CSince $h(1)=16$ is divisible by $8$, $h(k+1)$ is divisible by $8$.
   • DSince $h(k+1)=9h(k)-56$ is smaller than $9h(k)$, it is divisible by $8$.

   (c)

   Show that $h(k+1)=9h(k)-56$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A sequence is defined by $u_1=1$ and $u_{n+1}=\frac{u_n}{1+u_n}$ for $n\ge1$.

   (a)

   Find $u_2$, $u_3$ and $u_4$, and conjecture a formula for $u_n$.

   [3 marks]

   (b)

   Prove by induction that $u_n=\frac1n$ for all positive integers $n$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $S_n=\sum_{r=1}^{n}r\times2^r=1\times2+2\times4+3\times8+\dots+n\times2^n$.

   (a)

   Prove by induction that $S_n=(n-1)2^{n+1}+2$ for all positive integers $n$.

   [6 marks]

   (b)

   Use the result in part (a) to find (i) $\sum_{r=6}^{10}r\times2^r$ and (ii) the least $n$ for which $S_n>10^6$.

   [6 marks]

   Total for question 8: 12 marks

End of questions