Recitation 11 immediately, she uses the solution described above to advance another 3/4 of a day into the desert and then returns home. Thus, she reaches of a days walk into the desert (d) How can the explorer go as far as possible is she withdraws n gallons of water? Express your answer in terms of the Harmonic number Hn, defined by 111 Hn 123 Solution. With n gallons of water, the explorer can reach a point Hn/2 days into the desert Suppose she makes n trips 1/(2n) days into the desert, dropping off(n-1)/n gal lons each time. Before she leaves the cache for the last time, she has n gallons plus enough for the walk home. So she applies her(n- 1)-day strategy to go an addi- tional Hn-1/2 days into the desert and then returns home. Her maximum distance from the oasis is then: 1 H (e) Use the fact that Hn wInn to approximate your previous answer in terms of logarithms Solution. An approximate answer is(In n)/2 (f)Suppose that the shrine is d=10 days walk into the desert. Relying on your ap proximate answer, how many days must the explorer travel to recover the Holy Grail? Solution. She obtains the grail when This requires about n≥e20=4.8·10 days.Recitation 11 2 immediately, she uses the solution described above to advance another 3/4 of a day into the desert and then returns home. Thus, she reaches 1 1 1 11 + + = 6 4 2 12 of a days’ walk into the desert. (d) How can the explorer go as far as possible is she withdraws n gallons of water? Express your answer in terms of the Harmonic number Hn, defined by: 1 1 1 1 Hn = + + 1 2 3 + · · · n Solution. With n gallons of water, the explorer can reach a point Hn/2 days into the desert. Suppose she makes n trips 1/(2n) days into the desert, dropping off (n − 1)/n gal￾lons each time. Before she leaves the cache for the last time, she has n gallons plus enough for the walk home. So she applies her (n − 1)­day strategy to go an addi￾tional Hn−1/2 days into the desert and then returns home. Her maximum distance from the oasis is then: 1 + Hn−1 = Hn 2n 2 2 (e) Use the fact that Hn ∼ ln n to approximate your previous answer in terms of logarithms. Solution. An approximate answer is (ln n)/2. (f) Suppose that the shrine is d = 10 days walk into the desert. Relying on your ap￾proximate answer, how many days must the explorer travel to recover the Holy Grail? Solution. She obtains the Grail when: Hn ln n ≥ 10 2 ≈ 2 This requires about n ≥ e20 = 4.8 · 108 days