Using Principle of mathematical induction prove that $6^{n} - 1$ divisible by $5$ .

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Solution

$\begin{matrix} P (n) = 6^{n} - 1 P (1) = 6^{1} - 1 = 5 i s d i v i s i b l e b y 5 L e t, P (k) i s d i v i s i b l e b y 5 6^{n} - 1 = 5 m . . . . . . (1) P (k + 1) = 6^{k + 1} - 1 = {6.6}^{k} - 1 = (5 + 1) {.6}^{k} - 1 = 5 \times 6^{k} + 6^{k} - 1 = 5 \times 6^{k} + 5 m = 5 (6^{k} + m) P (k + 1) i s d i v i s i b l e b y 5 S o, 6^{n} - 1 i s a l w a y s d i v i s i b l e b y 5. \end{matrix}$

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