The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 5\) and \(u_{n+1} = 6u_n + 5\) for \(n \geq 1\).

(a) Prove by induction that \(u_n = 6^n - 1\) for all positive integers \(n\).

(b) Deduce that \(u_{2n}\) is divisible by \(u_n\) for \(n \geq 1\).

The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 5\) and \(u_{n+1} = 6u_n + 5\) for \(n \geq 1\).

(a) Prove by induction that \(u_n = 6^n - 1\) for all positive integers \(n\).

(b) Deduce that \(u_{2n}\) is divisible by \(u_n\) for \(n \geq 1\).

Prove by mathematical induction that \(2025^n + 47^n - 2\) is divisible by 46 for all positive integers \(n\).

Prove by mathematical induction that \(6^{4n} + 38^n - 2\) is divisible by 74 for all positive integers \(n\). [6]

Prove by mathematical induction that \(6^{4n} + 38^n - 2\) is divisible by 74 for all positive integers \(n\).

Prove by mathematical induction that, for all positive integers n, \(5^{3n} + 32^n - 33\) is divisible by 31.

Prove by mathematical induction that, for all positive integers n, \(7^{2n} + 97^n - 50\) is divisible by 48. [6]

Prove by mathematical induction that \(2^{4n} + 3^{1n} - 2\) is divisible by 15 for all positive integers \(n\).

1 Prove by mathematical induction that \(3^{3 n}-1\) is divisible by 13 for every positive integer \(n\).

It is given that \(\mathrm{f}(n)=2^{3 n}+8^{n-1}\). By simplifying \(\mathrm{f}(k)+\mathrm{f}(k+1)\), or otherwise, prove by mathematical induction that \(\mathrm{f}(n)\) is divisible by 9 for every positive integer \(n\).

Prove, by mathematical induction, that \(5^{n}+3\) is divisible by 4 for all non-negative integers \(n\).

It is given that \(\phi(n)=5^{n}(4 n+1)-1\), for \(n=1,2,3, \ldots\). Prove, by mathematical induction, that \(\phi(n)\) is divisible by 8 , for every positive integer \(n\).

Prove by mathematical induction that, for all non-negative integers \(n\),
\(11^{2 n}+25^{n}+22\)
is divisible by 24 .

Prove by mathematical induction that, for all positive integers \(n, 10^{n}+3 \times 4^{n+2}+5\) is divisible by 9 .

Prove by mathematical induction that \(5^{2 n}-1\) is divisible by 8 for every positive integer \(n\).

It is given that \(\mathrm{f}(n)=3^{3 n}+6^{n-1}\).
(i) Show that \(\mathrm{f}(n+1)+\mathrm{f}(n)=28\left(3^{3 n}\right)+7\left(6^{n-1}\right)\).
(ii) Hence, or otherwise, prove by mathematical induction that \(\mathrm{f}(n)\) is divisible by 7 for every positive integer \(n\).

Prove by mathematical induction that, for all non-negative integers \(n\),
\(11^{2 n}+25^{n}+22\)
is divisible by 24 .