Consider this recursive definition of a set S:
(1, 5) âˆˆ S and (3, 7) âˆˆ S
If (a, b) âˆˆ S then (3a + 1, 3b âˆ’ 7) âˆˆ S
If (a, b) âˆˆ S and (c, d) âˆˆ S then (2a + d, 2b + c) âˆˆ S
Prove by structural induction that for any (x, y) âˆˆ S, x + 4 = y
Based on the result from Part 1, what can we say about (1000, 1004)? Briefly explain.
Show your work for all parts.
Solve the following recurrence together with the given initial conditions:
an = an-1 + 1/4 an-2; a0 = 0, a1 = 3
Consider the following sequence:
an = 2n + 3n + 4n
We can express this sequence as a recurrence of the form: an =c1 an-1+ c2 an-2+ c3an-3
What are the values of c1, c2, and c3?
What initial conditions do we need for the recurrence?
In attached file.