R Proof. The proof is by induction on n. Let P(n) be the proposition that Tn=2n-1. Base case: P(1)is true because T1=1=2-1 Inductive step: Now we assume Tn=2n-1 to prove that T+1=2n+I-l, wherein>1 2Tn+1 2(2-1) 2n+1-1 The first equality is the recurrence relation, and the second equation follows by the as- sumption P(n). The last step is simplification Our guess is now verified. This shows, for example, that the 7-disk puzzle will require 2-1=127 moves to complete. We can also now resolve our remaining questions about the 64-disk puzzle. Since T64=204-l, the monks must complete more than 18 billion billion steps before the world ends. Better study for the final 2 Graduate Student Job Prospects In a new academic field (say, computer science), there are only so many faculty positions available in all the universities of the world. Initially, there were not enough qualified candidates, and many positions were unfilled. But, over time, new graduates are filling the positions, making job prospects for later computer science students ever more bleak Worse, the increasing number of professors are able to train an increasing number of graduate students, causing positions to fill ever more rapidly. Eventually, the universities will be saturated; new computer science graduates will have no chance at an academic career. Our problem is to determine when the universities will stop hiring new computer science faculty and in particular, to answer the question, Are the 6.042 TAs doomed? Here are the details of the problem There are a total of N faculty positions available worldwide. This number never due to budgetary constraints Congress has passed a law forbidding forced retirement in universities, and no one will retire voluntarily.(This is true and a problem! ) In our analysis, therefore, once a faculty position is filled, it never opens up again Each year, every professor trains exactly 1 student who will go on to become a pro- fessor the following year. The only exception is that first year professors do not train students; they are too busy publishing teaching, getting grants, and serving on committees In year 0, there are no computer science professors in the world. In year 1, the first professor is hiredRecurrences 5 n Proof. The proof is by induction on n. Let P(n) be the proposition that Tn = 2 − 1. 1 Base case: P(1) is true because T1 = 1 = 2 − 1. Inductive step: Now we assume Tn = 2n − 1 to prove that Tn+1 = 2n+1 − 1, where n ≥ 1. Tn+1 = 2Tn + 1 = 2(2n − 1) + 1 = 2n+1 − 1 The first equality is the recurrence relation, and the second equation follows by the as￾sumption P(n). The last step is simplification. Our guess is now verified. This shows, for example, that the 7­disk puzzle will require 27 − 1 = 127 moves to complete. We can also now resolve our remaining questions about the 64­disk puzzle. Since T64 = 264 − 1, the monks must complete more than 18 billion billion steps before the world ends. Better study for the final. 2 Graduate Student Job Prospects In a new academic field (say, computer science), there are only so many faculty positions available in all the universities of the world. Initially, there were not enough qualified candidates, and many positions were unfilled. But, over time, new graduates are filling the positions, making job prospects for later computer science students ever more bleak. Worse, the increasing number of professors are able to train an increasing number of graduate students, causing positions to fill ever more rapidly. Eventually, the universities will be saturated; new computer science graduates will have no chance at an academic career. Our problem is to determine when the universities will stop hiring new computer science faculty and, in particular, to answer the question, “Are the 6.042 TAs doomed?” Here are the details of the problem. • There are a total of N faculty positions available worldwide. This number never changes due to budgetary constraints. • Congress has passed a law forbidding forced retirement in universities, and no one will retire voluntarily. (This is true and a problem!) In our analysis, therefore, once a faculty position is filled, it never opens up again. • Each year, every professor trains exactly 1 student who will go on to become a pro￾fessor the following year. The only exception is that first year professors do not train students; they are too busy publishing, teaching, getting grants, and serving on committees. • In year 0, there are no computer science professors in the world. In year 1, the first professor is hired