Induction for series and sequencesAQA A-Level Further Maths: Subtopic test

1. 1
   Let $S_n=\sum_{r=1}^{n}r(r+1)$ for positive integers $n$. A student is proving by induction that $S_n=\frac{n(n+1)(n+2)}{3}$.

   (a)

   Find the value of $S_4$.

   [1 mark]

   • A$20$
   • B$40$
   • C$30$
   • D$120$

   (b)

   Assume the result is true for $n=k$. Which expression is $S_{k+1}$ before any simplification?

   [1 mark]

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

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For positive integers $n$, let $T_n=\sum_{r=1}^{n}\frac{1}{r(r+1)}$. It is to be proved by induction that $T_n=\frac{n}{n+1}$.

   (a)

   Find the value of $T_3$.

   [1 mark]

   • A$\frac{11}{12}$
   • B$\frac{1}{12}$
   • C$\frac34$
   • D$\frac43$

   (b)

   Assume the result is true for $n=k$. Which expression equals $\frac{k}{k+1}+\frac{1}{(k+1)(k+2)}$?

   [1 mark]

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

   (c)

   Write down the basis step for $n=1$ and state the inductive hypothesis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The sequence $u_n$ is defined by $u_1=3$ and $u_{n+1}=3u_n-2$ for $n\ge1$. A student suggests that $u_n=2\times3^{n-1}+1$.

   (a)

   Find $u_2$, $u_3$ and $u_4$, and show that the suggested formula gives the correct value of $u_4$.

   [3 marks]

   (b)

   Prove by induction that $u_n=2\times3^{n-1}+1$ for all positive integers $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For positive integers $n$, let $A_n=\sum_{r=1}^{n}r^2$ and $B_n=\sum_{r=1}^{n}r^3$.

   (a)

   Prove by induction that $A_n=\frac16n(n+1)(2n+1)$ for all positive integers $n$.

   [6 marks]

   (b)

   Prove by induction that $B_n=\frac14n^2(n+1)^2$ for all positive integers $n$.

   [6 marks]

   Total for question 4: 12 marks

End of questions