List the first 10 lucas numbers and the first ten fibonacci numbers and then prove each of the following propositions. the second principle of mathematical induction may be needed to prove some of these propositions. ? (a) for each natural number n, ln d 2fnc1 fn. (b) for each n 2 n with n 2, 5fn d ln1 c lnc1. (c) for each n 2 n with n 3, ln d fnc2