Induction for series and sequencesAQA A-Level Further Maths: Mind map

Exam questions on Induction for series and sequences

1. Let $S_n=\sum_{r=1}^{n}r(r+1)$ for positive integers $n$. A student is proving by induction that $S_n=\frac{n(n+1)(n+2)}{3}$.
   Show that $\frac{k(k+1)(k+2)}{3}+(k+1)(k+2)=\frac{(k+1)(k+2)(k+3)}{3}$.2 marks
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 $T_n=\frac{n}{n+1}$.
   Write down the basis step for $n=1$ and state the inductive hypothesis.2 marks
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 $u_n=2\times3^{n-1}+1$.
   Find $u_2$, $u_3$ and $u_4$, and show that the suggested formula gives the correct value of $u_4$.3 marks