I solved three mathematical induction questions. I am not sure if they make sense. I have a long answer for question 5, and I am not confident about the yellow sentences in the solution.

Could you check them?

Thank you,

5. (10pts) Use Mathematical Induction to prove that every amount of postage of 8 cents or more can be formed using just 3-cent and 5-cent stamps. P 6. (10pts) Use Mathematical Induction to prove that for every integer n 2 1, 1 1 - 3 .5 . ... . (2n - 1) 2n 2 - 4 - 6 . ... . 2n 7. (10pts) Use Mathmatical Induction to prove that for every integer n 2 1, in Cji = (n-1)20+ + 2 i=15. Let P (n) be the statement that postage of n cents can be formed using 3 cent and 5 cent stamps. Basis Step: Postage of 8 cents can be formed using one 3 cent stamp and one 5 cent stamp. L Inductive Step: The inductive hypothesis is the statement that P (k) is true, so under the hypothesis, postage of k cents can be formed using 3 cent and 5 cent stamps. We assume that P (k) is true, then P (k + 1) is also true where k 2 8, so we need to show that if we can form postage of k cents, then we can form postage of k + 1 cents. We consider two cases wen at that at least one 3 cent stamp has been used and when no 3 cent stamps have been used. First, suppose that at least one 3-cent stamps were used to form postage of k cents. Then we can replace three 3-cent stamps with at two 5-cent stamps to form postage of k + 1 cents. But if no 3- cent stamps were used, we can form postage of k cents using only 5-cent stamps. Moreover, because k 2 8, we needed at least two 5-cent stamps to form postage of k cents. We can replace four 5-cent stamps with seven 3-cent stamps to form postage of k + 1 cents. This completes the inductive step. Because we have completed the basis step and the inductive step, we know that P(n) is true for all n 2 8. That is, we can form postage of n cents, where n 2 8 using just 3-cent and 5-cent stamps. This completes the proof by mathematical induction. 6. Basis Step: When n = 2, =