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where xh(n) is the amount borrowed in the nth month, and x,(n) is the amount paid in the nth month. Combining terns, we have y(n) - vy(n - 1) = xh(n) - x,(n) = x(n) where v = 1+B = 1+O. 10/12, andx(n) is the net amount of money in the nth month that leaves the bank. Because a principal of p dollars is borrowed during month zero, and payments of d dollars begin with month 1, the input x(n) is

In the real world, the government may change its spending and taxing policies for economic reasons or for purely political reasons.

We now want to find the value of d so that the mortgage is retired after 060 equal monthly payments. That is, we want to find d so that I y(360) = -[(p d - pv)v'60 - dl = 0 I-v

A generalized Fibonacci sequence is a sequence of numbers, x(n), that satisfies the difference equation

x(n+2)=x(n)+x(n+I)

That is, x ( n ) is the sum of the two previous values. The classical Fibonacci sequence results when the initial conditions are x(0) = 0 and x(1) = 1 . The Fibonacci numbers occur in such unsuspecting places as the number of ancestors in succeeding generations of the male bee, the input impedance of a resistor ladder network, and the spacing of buds on the branch of a tree.

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- 54 -

(a) Find a closed-form expression for x(n).

(h) Show that the ratio x ( n ) / x ( n 1) approaches the limit 2 / ( 1 8) + co.This ratio is known as n as the golden mean and was said by the ancient Greeks to be the ratio of the sides of the rectangle that has the most pleasing proportions.

(c) Show that the Fibonacci sequence has the following properties:

(a) Here we have a second-order linear constant coefficient difference equation that we want to solve. Let us begin by rewriting it in a slightly different form. Specifically, consider the following

HPFS, NTFS, and DFS. The Microsoft Distributed File System (DFS) provides a means of organizing directory shares on multiple computers into a single directory tree. Print devices with an associated printer driver are represented as virtual printer objects and referenced by a queue name. Multiple physical print devices, called a printer pool, may be referenced by a single queue name. The Local Security Authority (LSA) and Net Login service are responsible for verifying access rights to resources and authenticating login requests respectively. Resource permissions are definedby Access Control Lists (ACLs). Windows 2000 enhances the SAM/LSA mechanism by integrating Kerberos 5.0 for network authentication.

A discretionary scal action has a multiplier effect upon the equilibrium level of output. The size of the multiplier depends upon whether there is a change in government spending, or in net lump-sum tax revenues, and there is an income tax. The value for the multiplier for the change in government spending is Y/ G, while the value of the multiplier for the change in net lump-sum tax revenues is Y/ T. When there is no income tax, a change in government spending has the same multiplier effect [k = 1/(1 MPC)] as does a similar change in investment spending or net exports. The multiplier is smaller for changes in net lumpsum tax revenues; the tax multiplier kt = MPC(k) or MPC/(1 MPC) is for an economy with no income tax. An income tax reduces the value of both the expenditure and the lump-sum tax revenue multiplier since the amount of taxes paid to government is directly related to income earned. For example, when the income tax rate is 20 percent and personal income increases $10, tax payments to the government rise $2 and personal disposable income increases $8 rather than $10. Thus, an increase in personal income results in smaller increments in induced consumption, and therefore results in a smaller multiplied effect. When there is an income tax, the equation for the expenditure multiplier is k = 1/[1 MPC + MPC(t)], where t is the income tax rate. The equation for the lump-sum tax multiplier is kt = MPC(k) or MPC/[1 MPC + MPC(t)].

where we assume that x(n) = 0 for n < 0 (i.e, initial rest). Written in this form with the delayed unit sample on the right-hand side, we note that x(0) = 0 and .r(l) = I as desired and x(n 2) = .r(n) x(n 1) for n > 0. The solution to this difference equation may be found using z-transforms as follows:

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