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    Homework 6#

    Due: Thursday, October 8, 2026 23:59
    Submit: Gradescope, through Blackboard’s assignment area.

    Please write your homework by hand. Start each exercise on a new page, scan your work to a legible PDF, and clearly mark the pages associated with every exercise. Show the algebra used to obtain each generating function or closed formula.

    1. (From a generating function to a recurrence)#

    Points: 10

    The ordinary generating function of a sequence is

    1. (A) 4 points. Determine initial data , , and . Show your calculations. Explain which terms you disregard.

    2. (B) 6 points. Determine the recurrence satisfied by , including its valid index range, and use it to compute and . Explain your derivation.

    2. (From a recurrence to a generating function)#

    Points: 10

    Let be defined by Let .

    1. (A) 10 points. Multiply the recurrence by and sum over all . Rewrite every sum in terms of and the standard series Solve the resulting equation for and give your answer as a single rational function.

    3. (Solving a homogeneous recurrence)#

    Points: 10

    Solve the recurrence Your solution must include the characteristic polynomial, its roots, the general form of a solution to the recurrence, and the use of the two initial conditions to determine the constants. Check your closed formula at against a direct recurrence calculation.

    4. (A nonhomogeneous recurrence with two driving terms)#

    Points: 10

    Solve the recurrence

    1. (A) 4 points. Treat and as separate driving terms. For each driving term, write an appropriate trial form for a particular solution. Explain why the trial for needs an extra factor of .

    2. (B) 4 points. Substitute both trial forms into the recurrence and determine their unknown coefficients. Add the two particular solutions to the general homogeneous solution.

    3. (C) 2 points. Write down the general formula for the homogeneous solution.

    4. (D) 4 points. Use the initial conditions to determine the remaining constants. Check your final formula at against a direct recurrence calculation.

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