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Exercise 4. Let P (n) be the statement that a postage of n cents can be...

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Advance Math


Exercise 4. Let P (n) be the statement that a postage of ncents can be formed using just 4-cent stamps and 7-centstamps.
The parts of this exercise outline a strong induction proofthat P(n)is true for n?18.
a) Show statements P(18), P(19), P(20), and P(21) are true,completing the basis step of the proof.
b) What is the inductive hypothesis of the proof?
c) What do you need to prove in the inductive step?
d) Complete the inductive step for k ? 21.
e) Explain why these steps show that this statement is truewhenever n ? 18.
Exercise 5. Use strong induction to show that every positiveinteger n can be written as a sum of distinct powers of two, thatis, as a sum of a subset of the integers 20 =1,21 =2,22 =4, and soon. [Hint: For the inductive step, separately con- sider the casewhere k + 1 is even and where it is odd. When it is even, note that(k + 1)/2 is an integer.]

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