Prove each of the following statements using strong induction. (a) Prove that any amount of postage worth 8 cents or more can be made from 3-cent or 5-cent stamps. (b) Prove that any amount of postage worth 24 cents or more can be made from 7-cent or 5-cent stamps. (c) Prove that any amount of postage worth 12 cents or more can be made from 3-cent or 7-cent stamps.\

Prove that 3: 8 is equivalent to 12: 32.a. use diagrams to support your answer.

Pepe and leo deposits money into their savings account at the end of the month the table shows the account balances. if there pattern of savings continue and neither earns interest nor withdraw any of the money , how will the balance compare after a very long time ?

Carla was trying to decide what to do saturday evening. she has $100 available to spend for the evening. she has narrowed her choices down to these events: braves game, a concert of an artist she likes, or the georgia aquarium. admission to the braves game is $25, hot dogs cost $10, sodas cost $5. the concert costs $25 for admission, $20 for a souvernir t-shirt, and $10 for a soda. the georgia aquarium has the show of a lifetime with an all you can eat and drink that costs $100. carla does not want to take any driving chances, so in addition to the cost of the aquarium, she will also need to pay for her uber/lyft, which is $15 each way. which option should carla choose if she wants to spend the least? based on that choice, what then is carla's oppotunity cost? fully explain your answer in at least one paragraph.