3.6-60 Spring 2006 ===== Part (a). Intuition to solve the problem can be obtained by writing out explicitly the determinants B1, B2, B3, B4. Subject B3, B4 to cofactor expansion to get the recursion identity of 3.6-60a. The work sumbitted must be for dimension n, for full credit. Examples are not a proof of part (a). Use "..." notation to write out B_{n}, B_{n-1} and B_{n-2}. Cite steps where cofactor expansion by row and by column are used. Part (b). Induction is explained in the appendix to the Varberg calculus text currently used for Calc I,II,III. Part 3.6-60b is expected to be solved using the theory of mathematical induction. Graders will award two possible grades: 80, 100. For 100 percent, the work must use induction methods. For 80 percent, some drastic unforgivable error has to occur, for example, submitting a work which does examples but no induction, or choosing a sequence of details that bears little or no relationship to the expected details. Proof details (b) ========= While many possible definitions exist for statement P_n in math induction, all of them have to have two equations in order apply the recursion formula of (a): (a) B_k = 2 B_{k-1} - B_{k-2} for k > 3 The problem is that to use (a), two equations are needed for expansion of the RHS. Bear in mind that P_n = { B_n = n+1} does not work for two reasons: first, P_1 = { B_1=2} and B_1 is not defined, only B_2 and higher; second, not enough equations appear in P_n to expand the RHS of (a). Something like P_n = { B_{n+1} = n+2, B_{n+2} = n+3 } does work. The details do not appear here, but are expected on submitted work.