Problems: Section 5.2

The purpose of this exercise set is to have you practice setting up difference equations. Even where a problem seems to ask for an explicit solution (e.g., [16]), it suffices to set up the appropriate difference equation and initial conditions. If you can guess an explicit formula from your equation, great; if you can prove it, super; but neither is part of the assignment yet.

1.	Devise a difference equation for the series 1+4+9+ + n ².
2.	Consider the sequence 1, 3, 7, 13, 21, 31, . Devise a difference equation for the sequence. Devise a difference equation for the associated series.
3.	An arithmetic-geometric sequence is one that starts at some value a, and thereafter each term is obtained from the previous one by first multiplying by a constant ratio r and then adding a constant difference d. Find a difference equation for this sort of sequence.
4.	In Example 3, what recurrence do you get if you use k=n ?1 in Eq. (2)? Also, find a recurrence relating E ₂ _n to E _n by replacing n by 2 n and k by n.
5.	Modify the second equation of the national income model, Eq. (6), so that C _n =cY _n. That is, we now think of personal income as being consumed instantaneously, at least relative to the rate at which C + I + G

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Discrete Algorithmic Mathematics, Third Edition

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Problems: Section 5.2

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