6.042/18.] Mathematics for Computer Science March 15, 2005 Srini devadas and Eric Lehman Problem set 6 Solutions Due: Monday, March 28 at 9 PM Problem 1. Sammy the Shark is a financial service provider who offers loans on the fol- lowing terms Sammy loans a client m dollars in the morning. This puts the client m dollars in debt to sammy. Each evening, Sammy first charges a"service fee", which increases the clients debt y f dollars, and then Sammy charges interest, which multiplies the debt by a factor of p. For example, if Sammys interest rate were a modest 5% per day, then p would be1.05 (a)What is the client's debt at the end of the first day Solution. At the end of the first day, the client owes Sammy(m f)p=mp+ fp doll ars (b) What is the clients debt at the end of the second day? Solution((m+ f)p+ fp= mp2+ fp2+ fp (c) Write a formula for the client's debt after d days and find an equivalent clos orm Solution. The clients debt after three days is (((m+ f)p+ fp+f)p=mp'+fp+fp+ fp generalizing from this pattern the client owes k=1 dollars after d days. Applying the formula for a geometric sum gives mp+/./pd+ Problem 2. Find closed-form expressions equal to the following sums. Show your work� � � 6.042/18.062J Mathematics for Computer Science March 15, 2005 Srini Devadas and Eric Lehman Problem Set 6 Solutions Due: Monday, March 28 at 9 PM Problem 1. Sammy the Shark is a financial service provider who offers loans on the fol￾lowing terms. • Sammy loans a client m dollars in the morning. This puts the client m dollars in debt to Sammy. • Each evening, Sammy first charges a “service fee”, which increases the client’s debt by f dollars, and then Sammy charges interest, which multiplies the debt by a factor of p. For example, if Sammy’s interest rate were a modest 5% per day, then p would be 1.05. (a) What is the client’s debt at the end of the first day? Solution. At the end of the first day, the client owes Sammy (m + f)p = mp + fp dollars. (b) What is the client’s debt at the end of the second day? Solution. ((m + f)p + f)p = mp2 + fp2 + fp (c) Write a formula for the client’s debt after d days and find an equivalent closed form. Solution. The client’s debt after three days is (((m + f)p + f)p + f)p = mp 3 + fp 3 + fp 2 + fp. Generalizing from this pattern, the client owes d mp d + fp k k=1 dollars after d days. Applying the formula for a geometric sum gives: d+1 p − 1 − 1 mp d + f · p − 1 Problem 2. Find closed­form expressions equal to the following sums. Show your work