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\lhead{Proof by Induction - Divisibility}
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\rhead{\href{https://danieljsmith.org}{danieljsmith.org}}
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\begin{document}
\vspace*{20pt}
\tableofcontents
\newpage
\begin{enumerate}[
  label=\textbf{\arabic*.},
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% ---- Question 1 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q1]

Prove by induction that \[n^5-n\] is divisible by $5$ for all positive integers $n$ \marks{5}

\newpage

% ---- Question 2 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q2]

Prove by induction that \[9^n-8n-1\] is divisible by $64$ for all positive integers $n\in\mathbb{Z}^+$ \marks{7}

\newpage

% ---- Question 3 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q3]

Prove by induction that, for $n \in \mathbb{Z}^+$,
\[13^n+3\cdot 4^n+11\] is divisible by $12$ \marks{7}


\newpage

% ---- Question 4 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q4]

Prove by mathematical induction that, for all non-negative integers $n$,
\[
2^n+2\cdot 3^{2n}+4^{2n+1}
\]
is divisible by $7$ \marks{6}

\newpage

% ---- Question 5 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q5]

For each positive integer $n$, define

\[u_n=5^{2n}-2^n\]

Use proof by induction to show that $23$ is a factor of $u_n$ for all $n \in \mathbb{Z}^+$ \marks{6}

\newpage

% ---- Question 6 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q6]

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

\[
7^n-6n-1
\]

is divisible by $36$ \marks{6}

\newpage

% ---- Question 7 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q7]

The function $f$ is defined by
\[
f(n)=9^n-8n+63 
\] 
Prove by induction that $f(n)$ is divisible by $64$ for $n \geq 1$ \marks{6}


\newpage

% ---- Question 8 ----
\questionitem
% [Source: _QBT__Proof_by_Induction___Divisibility, Q8]

Prove by mathematical induction that
\[
7^{2n}+8\times 13^n-9
\]
is divisible by $36$ for all positive integers $n$ \marks{6}


\end{enumerate}
\end{document}
