Induction problem: Company ships sandwiches in boxes of 3 or 5 each.

[dominikm]

Hey, i have a problem with a certain exercise : Company ships sandwiches in boxes of 3 or 5 each. Use induction to prove that every number of sandwiches above 7 can be shipped in only full boxes of 3 or 5.

So I started by Base Induction :
for n = 8 8 = 5*1 + 3*1 so it's true.

I can't figure out the Step Induction, but i started to write every number as 3*k + 5*l and what i observed is that every number above 7 can be written like this :
8,11,14.... 3n + 5 = 3(n+1) + 2 , n>=1
9,12,15.... 3(n+2) , n>=1
10,13,16... 3(n+2) + 1, n>=1

I'd really appreciate any hints on this matter as I am clueless. Sorry for any mistakes, as seen above English isn't my native language.

 

[ksdhart2]

It seems to me that what you've done is sufficient, just in a different form than you're used to seeing. "Classical" induction attempts to show that if some proposition holds for some k, it also holds for k + 1. Naturally, this method requires one base case. Here, you've proved that if the proposition (i.e. a certain number of sandwiches can be shipped) holds for some k, it also holds for k + 3. To support this, you've given the three required base cases (i.e. k = 8, k = 9, and k = 10).

 

[pka]

It seems to me that what you've done is sufficient, just in a different form than you're used to seeing. "Classical" induction attempts to show that if some proposition holds for some k, it also holds for k + 1. Naturally, this method requires one base case. Here, you've proved that if the proposition (i.e. a certain number of sandwiches can be shipped) holds for some k, it also holds for k + 3. To support this, you've given the three required base cases (i.e. k = 8, k = 9, and k = 10).

Using only boxes of three or five, we cannot ship one, two, four, or seven sandwiches. But from eight on we can.
Look at these combinations.
\(\displaystyle \begin{array}{*{20}{c}} 8&9&{10}&{11}&{12}&{13}&{14}&{15}\\ \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow \\ {3 + 5}&{3 \cdot 3}&{2 \cdot 5}&{2 \cdot 3 + 5}&{4 \cdot 3}&{3 + 2 \cdot 5}&{3 \cdot 3 + 5}&{3 \cdot 5} \end{array}\)

How do we ship twenty-two sandwiches?
How do we ship thirty-eight sandwiches?

 

[dominikm]

Still i need to use induction, so I'd really appreciate if someone could show me the way to properly conduct it.

 

[Denis]

Google "chicken nuggets math"