2

Tell me about this exercise, I try to solve it but it was confusing A bank gives 20\$ and 50\$. I must use mathematical induction so that the bank will create whatever amount of money bigger or equal to 40\$, that it is multiple to 10. Prove that for every natural number $nā‰„4$ there are $l,m$ Natural ,so that $10n=20l+50m$.

How i try solve it:

Basic step: n=1 
induction situation: n=k so 10k=20l+50m I name this (1)relation
Basic Induction :n=k+1 so 10(k+1)=20l+50m 
                          k+1=2l+5m
                          k=2l+5m-1 i name this (2) relation
In (1) relation i replace the k from (2) so i have
10(2l+5m-1 +1)=20l+50m 
10(2l+5m)=20l+50m
CiaPan
  • 13,049
ek.Sek
  • 497

1 Answers1

1

As you observed, all we need to show is that for each natural number $k\geq4$, there exists nonnegative integers $m,n$ so that $k=2m+5n$. First, note that if $k=2m+5n$, then $k+2=2(m+1)+5n$, hence if $k$ can be written in desired form, so can $k+2$. But obviously $k=4,5$ work. So by induction, all other integers work too.

To be concrete, all the odd numbers $\geq5$ work since $$\begin{split}5&=2\times0+5\times1\\7&=2\times1+5\times1\\9&=2\times2+5\times1\\&\vdots\\5+2n&=2\times n+5\times1\end{split}$$ and all even numbers $\geq 4$ work since $$\begin{split}4&=2\times2+5\times0\\6&=2\times3+5\times0\\8&=2\times4+5\times0\\&\vdots\\2n&=2\times n+5\times0.\end{split}$$

YiFan Tey
  • 17,431
  • 4
  • 28
  • 66
  • so ,what i have done ,is right ?cause i am not sure if is "the solution or not" what i did. ā€“ ek.Sek Jan 22 '20 at 01:37
  • @mainklain no, I don't think it's right. You need to be clear about what you're doing when attempting a proof by induction: first proving the proposition for a base case (here $k=4$), then proving that the proposition being true for a certain $k$ implies that it is true for $k+1$. You should make clear what you're doing for each step. ā€“ YiFan Tey Jan 22 '20 at 01:50
  • 1
    ok thank you :) ā€“ ek.Sek Jan 22 '20 at 01:52