Consider the following recurrence relation. - {²(0-1) + 0² if n=0 (a) Compute the first eight values of P(n). (Enter your answers as a comma-separated list of values ordered with respect to increasing n.) 1,2,6,15,31,56,92,141,205 eBook P(n) = (b) Analyze the sequences of differences. (Enter your answers as first differences: 4,9,16,25,36,49,64 second differences: X third differences: 5.7.9.11.13.15 2,2,2,2,2 X What does this suggest about the closed form solution? X This suggests that the closed form will have degree 3 X (c) Find a good candidate for a closed-form solution. 1+ n(n+1) (2n+1) P(n) = 6 (d) Prove that your candidate solution is the correct closed-form solution. (Induction on n.) Let f(n) = comma-separated list of values ordered with respect to increasing n.) Base Case: If n = 0, the recurrence relation says that P(0) = 1, and the formula says that f(0) = Inductive Hypothesis: Suppose as inductive hypothesis that P(k-1) = Inductive Step: Using the recurrence relation, P(K) = P(k-1) + k², by the second part of the recurrence relation + k², by inductive hypothesis so, by induction, P(n) = f(n) for all n ≥ 0. for some k > 0. so they match.

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Eight puzzle problem

The N-puzzle problem contains N tiles (N+1 including empty tiles) where the value of N can be 8, 15, 24, etc. The puzzle is divided into the square root of (N+1) rows and the square root of (N+1) columns. These problems have an initial state or configuration and a goal state or configuration. The puzzle can be solved by shifting the tiles one by one in the empty space. The objective is to place the numbers on the tiles to match the final state using the empty space. The other name for the N-puzzle problem is the sliding puzzle.