lean_workbook_plus_36961

Base Cases: Divisibility by 11 holds for $n = 0$ and $n = 1$ : $11$ divides $1 - 1 = 0$ , and also $25 - 3 = 22$ Inductive Step: If divisibility by 11 holds for $n - 1$ and for $n$ , it is true for $n + 1$ . Note that: $a^{n + 1} - b^{n + 1} = (a + b)(a^n - b^n) - ab (a^{n - 1} - b^{n - 1})$ Or, for this particular case, $a = 25$ and $b = 3$ : $25^{n + 1} - 3^{n + 1} = 28(25^n - 3^n) - 75 (25^{n - 1} - 3^{n - 1})$ But the two terms on the right are divisible by $11$ , so the LHS must also be divisible by $11$ .