Please teach me^^....

For n = 1, 5^1 - 3^1 - 2^1 = 5 - 3 - 2 = 0, which is divisible by 15.
Thus it is true for n = 1.

Assume that it is true for n = k, i.e.
5^(2k - 1) - 3^(2k - 1) - 2^(2k - 1) = 15M, where M is an integer.

For n = k+ 1, (here k is a positive integer)
5^[2(k + 1) -1] - 3^[2(k + 1) -1] - 2^[2(k + 1) -1]
= 5^(2k - 1 + 2) - 3^(2k - 1 + 2) - 2^(2k - 1 + 2)
= 5^(2k - 1) * 5^2 - 3^(2k - 1) * 3^2 - 2^(2k - 1) * 2^2
= 25 * 5^(2k - 1) - 9 * 3^(2k - 1) - 4 * 2^(2k - 1)
= 25 * 5^(2k - 1) - 5 * 3^(2k - 1) - 4 * 3^(2k - 1) - 4 * 2^(2k - 1)
= 25 * 5^(2k - 1) - 5 * 3^(2k - 1) - 4[3^(2k - 1) + 2^(2k - 1)]
= 25 * 5^(2k - 1) - 5 * 3^(2k - 1) - 4[5^(2k - 1) - 15M]
= 25 * 5^(2k - 1) - 5 * 3^(2k - 1) - 4 * 5^(2k - 1) + 60M
= 21 * 5^(2k - 1) - 5 * 3^(2k - 1) + 60M
= 7 * 3 * 5 * 5^(2k - 2) - 5 * 3 * 3^(2k - 2) + 60M
= 15 * 7 * 5^(2k - 2) - 15 * 3^(2k - 2) + 60M
= 15[7 * 5^(2k - 2) - 3^(2k - 2) + 4M], which is divisible by 15

Thus it is true for n = k + 1.

By the principle of mathematical induction, it is true for all positive integers.