Put S_n = 4ⁿ + 15n - 10

With n = 1, S₁ = 4¹ + 15.1 - 10 = 9 

Suppose n = k \(\ge1\), so S_k = 4^k + 15k - 10 ⋮ 9

So we must prove that: S_k+1 ⋮ 9

We have: S_k+1 = 4^k+1 + 15(k+1) - 10

                         = 4(4^k + 15k - 10) - 45k + 55 - 10

                         =  4S_k - 9(5k - 5)

But S_k ⋮ 9 => 4S_k ⋮ 9. The other side, 9(5k - 5) ⋮ 9

So S_k+1⋮ 9

So 4ⁿ + 15n - 10 ⋮ 9

Done