barcode generator for ssrs Substituting the second equation into the first, we have in Software

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Substituting the second equation into the first, we have
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which is an equation that defines how many B particles there are in the reactor at time n. Because there is one cu particle in the reactor at time n = 0, it follows that there are eight / particles at time n = 1. Therefore. the initial condition associated with B ( n ) is B(1) = 8, and this may be incorporated into the equation as follows:
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THE Z-TRANSFORM
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with B(n) = 0 for n < I. Using z-transforms, we may solve this equation for B(n) as follows:
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[CHAP 4
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Taking the inverse z-transform, we have
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Finally, because the number of u particles at time n is equal to the number of number of particles at time n = 100 is
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B particles at time (n - I),
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the total
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A $100,000 mortgage is to be paid off in 360 equal monthly payments of d dollars. Interest, compounded monthly. is charged at the rate of 10 percent per annum on the unpaid balance (e.g., after the first month the total debt equals $100,000 ~ $ 1 0 0 , 0 0 0 ) Find the payment d so that the mortgage is paid in full . after 30 years, leaving a net balance of zero.
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This is the same problem that was solved in Prob. 1.39. Here, however, we will use the z-transform to find the solution. The total unpaid balance at the end of the nth month, in the absence of any additional loans or payments, is equal to the unpaid balance in the previous month plus the interest charged on the unpaid balance for the previous month. Therefore, if y(n) is the balance at the end of the nth month,
where B is the interest charged on the unpaid balance. In addition, the balance must be adjusted by the amount of money leaving the bank into your pocket, which is simply the amount borrowed in the nth month and the amount paid to the bank in the nth month. Thus
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
and the difference equation for y(n) becomes
Expressing this difference equation in terms of z-transforms, we have
Solving for Y(z), we find
Taking the inverse z-transforms yields
CHAP. 41
THE >TRANSFORM
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
Solving for d, we have
With v = $ and p = 100,000 we have which is the same as we had previously calculated.
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.
(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
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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