MARKS Page four 3. A version of the following problem first appeared in print in the 16th Century. A frog and a toad fall to the bottom of a well that is 50 feet deep. Each day, the frog climbs 32 feet and then rests overnight. During the night, it slides down 2 3 of its height above the floor of the well. The toad climbs 13 feet each day before resting. Overnight, it slides down 1 4 of its height above the floor of the well. Their progress can be modelled by the recurrence relations: • +1 1 = +32 3 , n n f f f1 = 32 • +1 3 = +13 4 , n n t t t1 = 13 where fn and tn are the heights reached by the frog and the toad at the end of the nth day after falling in. (a) Calculate t2, the height of the toad at the end of the second day. (b) Determine whether or not either of them will eventually escape from the well. 1 5