Week_Five_Problems - n that do not have two consecutive 0's...

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Week Five Problems – Recurrence Relations Name 1______________________________ Name 2______________________________ Name 3______________________________ Group_______________________________ Work in groups of three (or two if necessary). Please write the solutions nicely on the back of this paper and continue on more paper – not “tetrads”. 1) For each of these tell the degree of the relation and whether or it is linear or nonlinear, homogeneous or inhomogeneous, has constant coefficients or not. Circle any terms which are non-linear, inhomogeneous, or do not have constant coefficients. a. b. c. d. e. 2) Find the solution of 3) Find the solution of 4) Find the general solution of solution of 5) A ternary string is a string that only contains the symbols 0, 1 and 2. Let be the number of ternary strings of length
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Unformatted text preview: n that do not have two consecutive 0's. Find a recurrence relation for and use the recurrence relation to findHint – think of how many “good” sequences begin with 1, how many with 2, and how many with 0 (which you will have to break into a few cases). 6) Solve the recurrence relation Hints – First find a particular solution. Then note the roots of the characteristic equation of the associated homogenous relation can be written as . Use that to show that the general solution to the homogeneous equation can be written as. Add the particular solution and match the initial conditions. 7) For the nonlinear equation and use Excel or a short computer program to find to three significant digits. What can you conjecture about for large n ? What changes if you start with...
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