(b) How many dog licenses will Mayville issue in 2009? (c) If the present trend continues, how many dog licenses can Mayville expect to issue after many years? 9.3 Linear Difference Equations with Constant Coefficients 493 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? 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? 30An automobile advertisement states that a new automobile can be purchased for $175 are to be made for 60 months and interest is charged at the rate of 1.075% how much does this car cost? month. If payments per month compounded monthly, per 31. Write a recurrence relation and initial conditions for the number s, of sequences of 1s and 2s having a sum or 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 2 3) be the vertices of a cycle (as defined in Section 4.2), and let c, denote the numbe of distinguishable ways to color these vertices with the colors red, yellow, blue, and green so that no adjace 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 S0+Sor +sSor +Sor", Sn the sum of the first n + 1 terms of a geometric progression with first term so and common ratio r1. (Hint: = 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 diffe equation Sn sequence so, S1, 82... satisfies the first-order linear difference equation s = asn-1+b, then a" (so + c) - c if a 1 if a 1 { Sn So+nb b for all nonnegative integers n, where с %3 а — 1 þe a sequence satisfying the first-order linear difference equation Sn=as-1+b +$2+ +s. Prove by mathe matical induction that