Question

1+4+4+4+4(1+4) NO 4) A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions. a) Rabbit pairs are not fertile during their first two months of life, but thereafter they give birth to four new male/female pairs at the end of every month. b) No deaths occur. For all integers n 2 1, let sy be the number of pairs of rabbits alive at the end of month. Let 80 = 1. Find a recurrence relation for so, 81, 82, ..., and carefully justify your answer. Sz=1th So all pairs ( First Month) Sist Pairs lend Of First Moram (not Fertile S2=1 Pairs (end of Secord month) (not Fertice) Pairs (Fraile adir 2 morms and give boom to 4 OHLAY) Sy = 1+4+4 paus Coff spruj bram to protes) Is = 1+4+4+4 Since initice Single Pars are not fertile daway the fliss The month The number of Posrs Staged the same for the firsd Two Sn=5n-JNO Thin the pars increse by a new pans Pror Penn molto Sn = Sn. +45 n-3) explanation so my. ShaBt 465) = See 8. No on the keys page in the resources module. 4

Answer 1

Soon not since the initial pair of rabbits are fertile during the first two months, the number of pairs of rabbits to main , at the end of first and second month. S,=1 Sz=1 The no. of pairs of rabbits alive at the end of month n is the number of pairs of rabbits a live at the end of the previous month noi increased by the number of rabbits born at the end of month n. Sn = Sort? The no. of rabbits born at the end of month n is upairs per pair of rabbits that was alive three months earlier (as the Eabbits are infertiles during their first two months) Sn=Snit 4 Sn-3