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Asked Jun 2, 2019

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**A) determine which amounts of postage can be formed using just 3 - cent and 10 - cent stamps.**

3,6,9,10,12,13,15,16,18,19,21,23,24,26,27,29,30,...,L*3+H*10

**C) prove your answer to (a) using strong induction. How does this inductive hypothesis in this proof differ from that in the inductive hypothesis for proof using mathematical induction?**

Base step:

need to show is T for 3,13,26

Statment of the Base Step:

3 = L*3+H*10 non-negative integers

13 = L*3+H*10 non-negative integers

26 = L*3+H*10 non-negative integers

Statment of the Base Step:

3 = 1*3+0*10

13 = 1*3+1*10

26 = 2*3+2*10

Statement of the inductive step

Proof of the inductive step

Invoke the principle of strong induction

Since the base step and the inductive step are true by the principle of strong induction all amounts of postage __________ can be obtained using 3 and 10 cent stamps

Step 1

(A)

We are given that the postage can be formed by using the stamps of just 3-cent and 10-cent only,

So such amount can be formed by all linear combinations, where *x* and *y* are nonnegative integers. So, possible amounts are:

3,6,9,10,12,13,15,16,18,19,20,21,22,……

Here, we can notice that the stamps having amount more than or equal to 18 cents can be formed by just 3-cent and 10-cent stamps.

Step 2

(C)

Now, we will prove the result by strong induction: That amount of postage of *c* cents can be formed by using just 3-cent and 10-cent stamps.

Let *P*(*c*) be the statement that, the amount of postage of *c* cents can be formed by using just 3-cent and 10-cent stamps.

**Basis step: **

Take c=18, c=19 and c=20

For P(18), we have

Step 3

Since, 18 cents can be formed by using six 3-cent So, P(1...

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