I reckon it's safe to assume n can be negative?.

This may be kind of flimsy, but............

Let n=0, then 5(0)+7(0)=0, true for n=0

Let n=1, then 1=5(3)+7(-2), true for n=1.

Let n=2, then 2=5(6)+7(-4), true for n=2.

Let n=3, then 3=5(9)+7(-6), true for n=3.

Let n=4, then 4=5(12)+7(-8), true for n=4.

See the pattern?.

So, we increase a by 3 and decrease b by 2

n+1=5(k+3)+7(p-2)=5k+15+7p-14=5k+7p+1

But 5k+7p = n. So the induction holds.