Proof by Induction (Edexcel A Level Further Maths: Core Pure): Exam Questions

Prove by induction that for all positive integers $n$,

$\mathrm{f}\left(n\right)=2^{3n+1}+3\left(5^{2n+1}\right)$

is divisible by 17

Prove by induction that for $n\in {\mathrm{\mathbb{Z}}}^{+}$

$\sum _{r=1}^n(3r+1)(r+2)=n(n+2)(n+3)$

Prove by induction that for all positive odd integers $n$

$\mathrm{f}\left(n\right)=4^n+5^n+6^n$

is divisible by 15

Prove by induction that for all positive integers $n,$

${\left(\begin{array}{cc}3& 0\\ 6& 1\end{array}\right)}^n=\left(\begin{array}{cc}{3}^n& 0\\ 3(3^n-1)& 1\end{array}\right)$

Prove by induction that for all positive integers $n$,

$\mathrm{f}\left(n\right)=9^n-2^n$

is divisible by 7.