Question

Proof that any multiple of a thousand greater or equal than 5000 can be expressed as the sum of some multiple of 5000 and some multiple of 2000

So, the proof is supposed to be by induction.
I've checked the base case, 5000= 5000*1+2000*0
I got the hipotesis
1000k = 5000a+2000b
But i can't figure out how to use that in order to prove the proposition for k+1.

Thanks.

Answer 1

we can prove this

Answer 2

assume the statement is true for n=k. so 1000k = 5000*a + 2000*b , for k>=5

Answer 3

then prove for n=k+1