Prove that \[ 5 \cdot 12^{n} + 8 \cdot 38^{n} \] is evenly divisible by 13 for all non-negative integer values of $n$.
Prove that \[ 3 \cdot 19^{n} + 7 \cdot 9^{n} \] is evenly divisible by 10 for all non-negative integer values of $n$.
Prove that \[ 2 \cdot 11^{n} + 3 \cdot 26^{n} \] is evenly divisible by 5 for all non-negative integer values of $n$.
Prove that \[ 5 \cdot 27^{n} + 2 \cdot 34^{n} \] is evenly divisible by 7 for all non-negative integer values of $n$.
Prove that \[ 4 \cdot 27^{n} + 9 \] is evenly divisible by 13 for all non-negative integer values of $n$.
Prove that the sum of the odd numbers 1 to n is a perfect square using induction.