Discrete Mathematics 2 Recurrence Relations For each of the following attempts t

Question

Discrete Mathematics

2 Recurrence Relations For each of the following attempts to define a sequence via a recurrence relation do this: (a) State if the formula defines a sequnce and if not then explain why; (b) If it is a sequence then write out its first ten elements; (c) State whether this sequence is increasing, non-decreasing, constant, non-increasing, de- creasing, or neither of the above. 1, ao = 0, ai = 0, a2 = 1; an-an-1-an-2 + an-3 for n > 3. 2. ao = 1, al = 1, a2 = 2; an-2an-1-an-2 for n 3. 3. (10=1; an-2@m/2] 4, ao = 0, al = 1; an-an+1 + an-1 for n 2. = 2alnaj for n21.

Explanation / Answer

We experts are allowed by Chegg to solve only 1 problem. So, I will be working out only problem 1 in depth and similarly you can try to solve the rest and still if you need help, you can repost your problem.

1. a0=0,a1=0,a2=1

an = an-1 - an-2 + an-3

(a) This defines a sequence

(b) The first 3 elements of the sequence is already given.

a3 = a2 - a1 + a0

a3 = 1 - 0 + 0 = 1

Similarly, a4 = 1 - 1 + 0 = 0

a5 = 0 - 1 + 1 = 0

a6 = 0 - 0 + 1 = 1

a7 = 1 - 0 + 0 = 1

a8 = 1 - 1 + 0 = 0

a9 = 0 - 1 + 1 = 0

These were the first 10 terms of the sequence.

(c) Neither of the above. Because the sequence values sometimes go up, sometimes go down. So, isn't either constant, or increasing or decreasing always.

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