Perform the calculations in Liu Hui's algorithm for deter- mining л to find S2n for n = 6, 12, 24, 48, and 96.
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- Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .a. Excluding the identity cipher, how many different translation ciphers are there using an alphabet of n "letters"? b. Excluding the identity cipher, how many different affine ciphers are there using an alphabet of n "letters," where n is a prime?13. Solve the recurrence relation. Given:a0=3 a1= 6 an= 6a n-1 + 7a n-2 Show your work in the space provided. a. Find c1 and c2 an= (6)a n-1 + (7)a n-2 c1= c2= b. Substitute c1 and c2 into the following equation: r^2- c1r - c2= 0 show your work here: c. Identify a, b, and c in the quadratic equation. a= b= c= d. Use the quadratic formula to find the two roots. Here is the quadratic formula: -b + Vb^2- 4ac/2a Show your work here: e. Substitute two roots, r1 and r2 into the equation an= a1r1^n+ a2r2^n Show your work here: f. Now substitute to find two equations, a0 and a1 Remember to use the equation you found from step e. show your work here: a0= 3 =a1= 6 = g. Add the two equations together find a1 and a2 a1= a2= h. What is the solution to the recurrence relations? an= i. Find the 10" term of the sequence, using the solution to the recurrence relation you just found. Show your work here: a10=
- 7. Show that the recurrence relation has a general solution of the form Xn = 2xn-1 + xn-2 Ăn = A2n + Bạn Is the recurrence relation a good way to compute x, from arbitrary initial values xo and x₁?Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in n!, same with (n-r)! and r!. Then cancel numbers appropriately in the numerator and denominator and simplify to one number.Assume that to lig in to a computer network a paaword must be entered. A hacker who is trying to break into the system rendomly types one password every 12 seconds. If the hacker does not enter a valid password within 6 minutes, the system will not allow any further attempts to log in. The password consists of any sequence of two letters and three digits. Cases does not matter for the letters. Thus, B12q5 and b12Q5 are considered the same password. What is the probability that the hacker will be successful in discovering a valid passward? ( Hint: If E is an event, then P(E')=1-P(E), where E' is the complement of event E.)
- Find the degree of the following Ihcc recurrences: 1. ат — ап-2 +4am-з + 2an-6 2. аn 5an-1 3. аn — ап-1 +ап-2 +an-3 4. аn = an-7 +7an-8A lake initially contains 1000 fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by 10% each month. However, factoring in all causes, 80 fish are lost each month. Give a recurrence relation for the population of fish after n months. How many fish are there after 5 months? If your fish model predicts a non-integer number of fish, round down to the next lower integer1. Suppose we build an LFSR machine that works mod 3 instead of mod 2. It uses a recurrence of length 2 of the form Xn+2 = CoXn+ c1xn+1__mod 3 to generate the sequence 1, 1, 0, 2, 2, 0, 1, 1. Set up and solve the matrix equation to find the coefficients co and c1.
- ] *CountAlg.java - Notepad e Edit Format View Help Ln 28, Col 1 100% Windows (CRLF) UTF-8 Problem 2. The Fibonacci numbers are defined by Fo = 0, F1 = 1, F = Fr + Fx-1 for k = 1, 2, .... Relations such as this are called recurrence relations or difference equations. We can recast this relation in "initial- value problem" form as follows: () - ( :) (A). (E) - () Fk+1 1 Fk k = 1, 2, ... k-1 It is apparent from this that (2)-G )" (E). k-1 F1 (A). Fk k = 2,3, .... Find F30 as follows: First find the eigenvalues and associated eigenvectors of A := (}); next, find P, P-1, and D that satisfy A = PDP-1; then raise the diagonal entries in D (the eigenvalues of A) to the 29th power; and finally, evaluate the first component of PD29 P-1 to obtain F30. 1 1:36 PM 0 Type here to search a Rain... 后 ) 4/26/2022 (1can you choose correct one for 2 of them (seperately)Find a recurrence relation for the number of bit strings of length n that contain the string 00 or the string 10. Be sure to include the required initial conditions.
