I need help with question 3 please. Both a and b. Thank you.

MATH 674.1 Written Assignment #2 Fall 2016 1. Tommy and Timmy Twintz want to establish a scholarship that makes two payments every year. They want the first payment of $5,000 to go to the very first freshman who arrives on campus. The second payment of $20,000 will be handed out six months later to the freshman with the highest first term average. These payments will be made forever. Assuming money can be invested at i = 5%, how much will the Twintzes need to fund this scholarship if the first payment is to be made six months from today? 1 MATH 674.1 Written Assignment #2 Fall 2016 2. Polly Promises turns 25 today. She has decided to give up what she considers her three biggest weaknesses: smoking, drinking coffee and buying lottery tickets. As a gift to herself, she further vows that all the money she saves will be deposited into her savings account, currently paying i(365) = 3%, and left there until she retires at age 65. Polly's habits had been smoking one pack of cigarettes a day, drinking three large coffees a day and buying four lottery tickets every week. Assuming Polly is successful in this newfound yet somewhat puritanical quest, how much money will she have in her savings at retirement if cigarettes cost $12 per pack, coffee costs $1.80 per cup and lottery tickets cost $5 each? 2 MATH 674.1 Written Assignment #2 Fall 2016 3. While trekking in Nepal, Jim becomes separated from his friends. He stumbles into a cave where an old monk offers him a cup of water. After drinking it, Jim feels energized and asks what was in the water. \"You have just drunk from the Holy Grail,\" replies the monk, \"now you will live forever.\" Jim is ecstatic. But after several days of drunken debauchery, the bartender demands to be paid. Jim realizes that if he is going to live forever, he will either have to work forever, or make money by investing. (A) Assuming Jim can earn 5% a year in the investment markets and he can manage to live on $50,000 annually (paid at the start of every year), how much money will Jim need to finance his now lengthened existence, if he depends solely on investment income? (B) Suppose after 50 years, Jim has actually earned 6% per year. If he is still living on $50,000 per year, how much will be in his investment account? 3 MATH 674.1 Written Assignment #2 Fall 2016 4. (BONUS) Assume i > 0. Let EAR(i, m) be the effective annual rate when the interest rate i is compounded m times per year. Show that i EAR(i, m) ei 1. 4