9.3 Linear Difference Equations with Constant Coefficients 493 How many dog licenses will Mayville issue in 2009? (b) If the present trend continues, how many dog licenses can Mayville expect to issue after many years? 27. Michelle has just opened a savings account with an initial deposit of $1000. From the money she earns from her part-time job, Michelle will add $100 to her savings account at the end of each month. If the account compounds interest monthly at the rate of 0.5% per month, how much will it be worth two years from now (c) 28. Suppose that a corporate executive deposits $2000 per year for 35 years into an individual retirement account. If interest is compounded annually at the rate of 8%, how much will the account be worth after the last deposit? 29, The Johnson family is considering the purchase of a house costing $159,000. They will make a $32,000 down payment and take a 30-year mortgage for the remainder of the cost. The mortgage compounds interest monthly at the rate of 0.9% per month. How much will the Johnsons' monthly payment be under these conditions? 30. An automobile advertisement states that a new automobile can be purchased for $175 per month. If payments are to be made for 60 months and interest is charged at the rate how much does this car cost? of 1.075% per month compounded monthly, 31. Write a recurrence relation and initial conditions for the number s, of sequences of 1s and 2s having a sum of n. Use these to obtain a formula expressing Sn as a function of n. 32. Suppose that a bank, in order to promote long-term saving by its customers, authorizes a new savings account that pays 6% interest on money during the first year it is in the account and 8.16% interest on money that is in the account for more than one year. Interest is to be compounded annually. If you deposit $1100 into such an account and allow the interest to accumulate, how much will the account be worth in n years? 33. Let v1, v2, . . . , Un (n > 3) be the vertices of a cycle (as defined in Section 4.2), and let c, denote the number of distinguishable ways to color these vertices with the colors red, yellow, blue, and green so that no adjacent vertices have the same color. Determine a formula expressing cn as a function of n. 34. Use Theorem 9.1 to find a formula for +Sor", = 50 +Sor + sor Sn the sum of the first n +1 terms of a geometric progression with first term so and common ratio r 1. (Hint: The sequence So, S1, S2... satisfies the first-order linear difference equation Sn = rsn-1 + So with initial term so.) 35. Prove by mathematical induction that if so, S1, S2, .. . is a sequence satisfying the first-order linear difference equation sn= asn-1+b, then a" (so + c) - c if a 1 if a = 1 { Sn S0+nb for all nonnegative integers n, where b а — 1 0 by tn= So +S1 +S2 +..+Sn. Prove by mathematical induction that a"+1- (n+1)a +n 36. Let so, S1, S2,... be a sequence satisfying the first-order linear difference equation s= ash-1 +b for n 1, and define t, for n if a #1 an+1 (a - 1)2 S0