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# Show that postage of 6 cents or more can be achieved by using only 2-cent and 7-cent stamps.

Solution

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To proof: A postage of at least 6 cents can be obtained by using only 2-cent and 7-cent stamps.

$\textbf{PROOF BY STRONG INDUCTION}$

Let $P(n)$ be the statement "A postage of $n$ cents can be obtained by using only 2-cent and 7-cent stamps".

$\textbf{Basis step}$ $n=6$

We obtain a 6 cent postage by using three 2-cent stamps.

$2\cdot 3=6$

We obtain a 7 cent postage by using one 7-cent stamps.

Thus $P(6)$ and $P(7)$ are true.

$\textbf{Inductive step}$ Let $P(6),P(7),P(8),....,P(k)$ be true while $k\geq 7$.

$\begin{matrix}\text{A postage of i cents can be obtained}\\ \text{ from 2-cent and 7-cent stamps for }i=6,7,...,k\end{matrix}$

We need to proof that $P(k+1)$ is true.

Since $k\geq 7$, we also know $k+1\geq 8$.

If we then use one 2-cent stamp, then we still require a postage of $k-1\geq 6$ cents.

Since $P(k-1)$ is true, $k-1$ cents can be obtained from 2-cent and 7-cent stamps and thus $k+1$ can be obtained from 2-cent and 7-cent stamps (by using one additional 2-cent stamp).

Thus $P(k+1)$ is true.

$\textbf{Conclusion}$ By the principle of strong induction, $P(n)$ is true for all natural numbers $n$.

$\square$

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