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these difference equations may be solved by induction to yield the following recurrence formula for A k ( z ) :

10 12 14 16 18 20 22 = = = = = = =

which is called the step-up recucrion. The recursion is initialized by setting A o ( z ) = 1. This recurrence formula also defines a recurrence relation for the coefficients a k ( i )of A k ( z ) ,which is

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In languages such as C++, such a run-time error normally causes the program to crash An array index can have type byte, char, short, or int The Array class cannot be instantiated because its constructor is declared private The simplest way to print an array of objects is to pass it to the ArraystoList() method which produces a List object that can be printed directly with the Systemoutprintln() method The binary search will probably not work unless the sequence is sorted first If the sequential search is applied to an element that occurs more than once in an array, it will return the index of the one that is closest to the beginning of the array If the binary search is applied to an element that occurs more than once in an array, it could return the index of any one of them.

SIGNALS AND SYSTEMS The output of this system at time n is the product of the values of the input x ( n ) at times n - 1, . . . , n Therefore, because the output depends only on past values of the input signal, the system is causal.

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2.25 Pierre de Fermat (1601 1665) conjectured that there are infinitely many prime numbers of p 2 the form n = 2 +1 for some integer p. These numbers are called Fermat primes. For exam1 2 ple, 5 is a Fermat prime because it is a prime number and it has the form 2 +1. Write a program that finds all the Fermat primes that are in the range of the int type. Use the Primes class from Problem 2.22 and the Math.pow() method. Your first 5 lines of output should look like this:

( f ) This system is not causal, which may be seen easily if we rewrite the system definition as follows:

It depends upon how close their indexes are to multiples of midpoints of subintervals For example, if the binary search is applied in an array of 10,000 elements, searching for a value that is repeated at locations 0 99, the search would return the index 77 on the 7th iteration..

Therefore, the input must be known for all n 5 0 to determine the output at time n . For example, to find y ( - 5 ) we must know x(O), x ( - I), x(-2), . . .. Thus, the system is noncausal.

2.26 Charles Babbage (1792 1871) obtained the first government grant in history when in 1823 he persuaded the British government to provide 1000 to build his difference engine. In his grant proposal, Babbage gave the formula x 2 + x + 41 as an example of a function that his computer would tabulate. This particular function was of interest to mathematicians because it produces an unusual number of prime numbers.Primes that have this form n = x 2 + x + 41 for some integer x could be called Babbage primes. Write a program that finds all the Babbage primes that are less than 10,000. Use the Primes class from Problem 2.22. Your first five lines of output should look like this:

Determine which of the following systems are stable:

2.4 2.5 2.6 2.7 2.8 2.9

(b) y ( n ) = ex(")/ x ( n - 1)

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