Let the statement be true for n = k

That means 3 * 5^(2k-1) + 2^(3k-2) = 17 p for some p in N

That gives 3 * 5^(2k-1) = 17 p - 2^(3k-2) ------- [ 1 ]

Now for n = k+1 we have

3 * 5^{2(k+1) -1} + 2^{3(k+1)-2} = 3 * 5^(2k+1) + 2^(3k+1) = 3 * 5^(2k-1+2) + 2^(3k-2+3)=3 * 25 *5^(2k-1) + 8*2^(3k-2)

= 25* { 17 p - 2^(3k-2)} + 8*2^(3k-2) = 25* 17 p - 25* 2^(3k-2) + 8*2^(3k-2)= 25* 17 p - 17* 2^(3k-2)

And you have the result