3. We found in class that if Cn counts the number of ways to make change (in standard coins) for n cents, the generating function for this sequence is 8 Σ cnx" n=0 = 1 (1 − x)(1 − x5) (1 − x¹⁰) (1 – x²5) · - Let an again be the number of ways to make change for n cents, but with the added feature that we care about the order in which the coins are chosen. We could say, for example, that an is the number of ways to feed coins totalling n cents one by one into a machine. Give a closed-form generating function for the sequence (an). [Hint: you will use 0 coins in total, or 1 coin in total, or 2 coins in total...]

Q: 313. Let fn stand for the nth term of the Fibonacci sequence. What is remarkable about the sequence…

A:

Q: Consider the sequence {n} defined by ₁ = = 3.1, T₂ = 3.14, T3 = 3.141,...; for each n, πn shows the…

A:

Q: 4. Define the sequence an by a, = 1, a, = 33, and an = 3an-1+ 10an-2 for n 2 3. Prove that an = 5" –…

A: Concept used :- we use mathematical induction to solve the given question.

Q: 12. Prove the following facts about the Fibonacci sequence (Fn): (1) (Chapter 10, #26) E-1 F =…

A: Given, The Fibonacci sequence (Fn) Fn=Fn-1+Fn-2, where F0=0 & F1=1

Q: 12. Suppose the sequence ao, aj, a2, a3, ... is defined as follows: ao = 0, aj = 1, and for n > 2,…

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Q: 1. Write out the next 4 terms of the sequence a = a,' +n-1; a, =1 (this means %3D а, а,, а,, а,).

A: Given a0=1 and an+1=an3+n-1a1 means n=0So, a1=a0+1=13+0-1          =0a2=a1+1=03+1+-1              =0

Q: 2. The first few terms of a sequence (x) are given below. Assuming that the “natural pattern’…

A: Here, we have to given first few terms of a sequence. Assuming that the natural pattern indicated by…

Q: 5. Find the 6th term of the recursive sequence defined by a₁ = 0, an+1=2a₁ +3.

A: