3. Strong Induction (11 points) (1) (6 points) Let P(n) be the statement that a postage of n cents can be formed using just 3-cent stamps and 7-cent stamps. The Induction and Recursion parts of this exercise outline a strong induction proof that P(n) is true for n 2 18. (a) (1 points) Show that P(12), P(13), and P(14) are true, which com- pletes the base case (b) (1 points) What is the inductive hypothesis? (c) (1 points) What do you need to prove in the inductive step? (d) (3 points) Complete the inductive step for k 2 14. (2) (5 points) Use strong induction to show that every positive integer can be written as a sum of distinct powers of two (ie, 20-12-2, 22- 4.2-8.21=16, . . .) For example: 19 16+2+1-21+21 +2 Hint: For the inductive step, separately consider the case where k +1 is even and where it is odd. When it is even, note that (k+1)/2 is an nteger 3. Strong Induction (11 points) (1) (6 points) Let P(n) be the statement that a postage of n cents can be formed using just 3-cent stamps and 7-cent stamps. The Induction and Recursion parts of this exercise outline a strong induction proof that P(n) is true for n 2 18. (a) (1 points) Show that P(12), P(13), and P(14) are true, which com- pletes the base case (b) (1 points) What is the inductive hypothesis? (c) (1 points) What do you need to prove in the inductive step? (d) (3 points) Complete the inductive step for k 2 14. (2) (5 points) Use strong induction to show that every positive integer can be written as a sum of distinct powers of two (ie, 20-12-2, 22- 4.2-8.21=16, . . .) For example: 19 16+2+1-21+21 +2 Hint: For the inductive step, separately consider the case where k +1 is even and where it is odd. When it is even, note that (k+1)/2 is an nteger