Question

Prove the statement n cents of postage can be formed using just 4-cent and 11-cent stamps using mathematical induction, where n ≥ 30.

Click and drag the given steps (on the right) to the corresponding step names given on the left) to carry out the inductive steps of the proof, after the inductive hypothesis has already been assumed in (b). Step 1 Replace eight 4-cent stamps by three 11-cent stamps, and we have formed k+ 1 cents in postage (3. 11 = 8.4 + 1). Because k 2 30, if there only 4-cent stamps, there must be at least eight. Step 2 Otherwise, k cents was formed from just 4-cent stamps. Step 3 Because k> 30, there must be at least eight 4-cent stamps involved. If k cents included an 11-cent stamp, then replace it by three 4-cent stamps (3.4 = 11 + 1), and we have formed k + 1 cents in postage. Step 4 Otherwise, k cents was formed from just two 4-cent stamps.

Answer 1

Base Case : 30 cents can be formed using two 11-cent stamps and two 4-cent stamps as 2*11+2*4 = 30

Inductive Hypothesis : If k > 30, then we can from a k-cent stamps using only 4-cent and 11-cent stamps.

The Inductive steps of the Proof are as follows :

Step 1 : If k cents included an 11-cent stamp, then replace it by three 4-cent stamps (3+4 = 11+1), and we have formed k+1 cents in postage.

Step 2 : Otherwise, k cents was formed from just 4-cent stamps.

Step 3 : Because k > 30 there must be at least eight 4-cents stamp involved.

Step 4 : Replace eight 4-cent stamps by three 11-cents stamps, and we have formed k+1 cents in postage (3*11 = 8*4+1)

We have covered all possible cases and we can show that from if k cents could be formed from 4-cent and 11-cent stamps then k+1 cents can be formed also. This completes our inductive proof.

Note that in Step 3 we have taken k>30 this is because we have tested our base case of 30 cents already. Also, we conclude that at least eight 4-cent stamps are required because k>30 is 31 (which requires at least one 11-cent stamp and as we are considering the case of formation using only 4-cent stamps, hence it does not count) and then 32 which requires eight 4-cent stamp.