Let p(n) be the statement that a postage of n cents can be formed using just 4-cent stamps and 7-cent stamps. the parts of this exercise outline a strong induction proof that p(n) is true for all integers n ≥ 18.
a) show that the statements p(18), p(19), p(20), and p(21) are true, completing the basis step of a proof by strong induction that p(n) is true for all integers n ≥ 18.
b) what is the inductive hypothesis of a proof by strong induction that p(n) is true for all integers n ≥ 18?
c) what do you need to prove in the inductive step of a proof that p(n) is true for all integers n ≥ 18?
d) complete the inductive step for k ≥ 21.
e) explain why these steps show that p(n) is true for all integers n ≥ 18.

(50 ) mrs. julien’s and mrs. castillejo’s classes are selling cookie dough for a school fundraiser. customers can buy packages of macadamia nut chip cookie dough and packages of triple chocolate cookie dough. mrs. julien’s class sold 25 packages of macadamia nut chip cookie dough and 30 packages of triple chocolate cookie dough for a total of $221.25. mrs. castillejo’s class sold 5 packages of macadamia nut chip cookie dough and 45 packages of triple chocolate cookie dough for a total of $191.25. (a) write the system of equations that model the problem. be sure to explain which equation represents which situation. (b) find the cost of each type of cookie. show your work. (c) explain which method you used to solve the system and why you chose that method.

Randy and trey take turns cleaning offices on the weekends. it takes randy at most 4 hours to clean the offices. it takes trey at most 6 hours to clean the offices. what is the greatest amount of time it would take them to clean the offices together? 2.4 hours 2.5 hours

Awavelength of 610 nm was found to be the most intense wavelength emitted by copper. answer the following questions about this wavelength of light. a) what is this wavelength in cm? b) what is the frequency that corresponds to this wavelength? c) what is the energy of light of this wavelength? d) what is the wavenumber corresponding to this wavelength?