Proof by mathematical inductionEdexcel International A Level Further Maths: Mind map

Exam questions on Proof by mathematical induction

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 $n=1$.
   Assuming the result is true for $n=k$, show that it is true for $n=k+1$.2 marks
2. Let $f(n)=7^n-1$ for positive integers $n$.
   Hence, or otherwise, show that if $f(k)$ is divisible by $6$ then $f(k+1)$ is divisible by $6$.2 marks
3. A sequence is defined by $u_1=2$ and $u_{n+1}=2u_n+3$ for $n\ge1$.
   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