Algorithm 1 PowerOfThree if n=0 then return 1 end if r+ 1 if n is odd then n+ n-1 rt 3 end if q+ PowerOfThree(n/2) return r x q a. Derive a recurrence relation T(n) that represents the running time of PowerOfThree. You are permitted to condense any constant time terms in the recurrence into the constant 1. b. Find a closed form for your recurrence from part (a). c. Prove (by induction) that your closed form is correct. d. Prove the following statement by induction. For all n > 0, the value of PowerOfThree(n) is equal to 3n This is a proof of correctness and is completely independent from your running time proofs in parts (a), (b) and (c). The proof should clearly establish that, for a particular value of n, the return value of the algorithm will always equal 3". Using strong induction (where the hypothesis is that the algorithm is correct for all values less than n) will likely help Algorithm 1 PowerOfThree if n=0 then return 1 end if r+ 1 if n is odd then n+ n-1 rt 3 end if q+ PowerOfThree(n/2) return r x q a. Derive a recurrence relation T(n) that represents the running time of PowerOfThree. You are permitted to condense any constant time terms in the recurrence into the constant 1. b. Find a closed form for your recurrence from part (a). c. Prove (by induction) that your closed form is correct. d. Prove the following statement by induction. For all n > 0, the value of PowerOfThree(n) is equal to 3n This is a proof of correctness and is completely independent from your running time proofs in parts (a), (b) and (c). The proof should clearly establish that, for a particular value of n, the return value of the algorithm will always equal 3". Using strong induction (where the hypothesis is that the algorithm is correct for all values less than n) will likely help