Question

which amount of money can be formed from 2dollar bill and 5 dollar bill?
Please, help

Answer 1

Um... $10

Answer 2

$20

Answer 3

2n+5m
n,m->I

Answer 4

Thank you, but I need to know the formula of n, where n = 2x+5y

Answer 5

what do you mean by formula for n?
isn't it just n=2x+5y where x and y are integers

Answer 6

20

Answer 7

\[x = \frac{ n - 5y }{ 2 }\]

Answer 8

2x+ 5y = n
So, the constraint is P(4) , since if n=4, then x =2,y =0

Answer 9

for n =5, then x = 0, y =1

Answer 10

But if n =3, you don't have P(3) because there is no x,y satisfy 2x+5y =3

Answer 11

P(4) and up checked
Now , by strong induction, we need to find the formula of n.

Answer 12

\(P(4)\cap P(5)\cap P(6)\cap \cdots \cap P(n)\rightarrow P(n+1)\)

Answer 13

\(P(n-3)P(n-2)P(n-1) = P(n) - stuck!!) = P(n+1)\)

Answer 14

And I need help to unlock the point I stuck. Just from there, I can construct the formula of n and prove it by induction.

Answer 15

I got it

Answer 16

Let P(n) := we can form all amounts of money greater or equal to 4 from 2 dollar bill and 5 dollar bill
P(4) = 2*(2) +0*(5)

P(5) =0*(2) +1*(5)

So base cases are true.

Answer 17

Hypothesis step: Suppose P(k) is true for k amount of money

so, we need prove P(k+1)

From HS, \(4\leq j\leq k\)

So P(k-1) is true. if P(k-1) is true, now we add one 2 dollar bill in that amount, we have k-1+2 = k+1, so the amount of k+1 is formed by 2 dollar bill and 5 dollar bill also.

Proof done.