By Abraham P Hillman

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We have seen that binomial coefficients, Fibonacci and Lucas numbers, and factorials may be defined inductively, that is, by giving their initial values and describing how to get new values from previous values. Similarly, one may define an arithmetic progression a1, a2, ... , t - 1. Then the values of a1 and d would determine the values of all the terms. A geometric progression b1, ... , bt is one for which there is a fixed number r such that bn+1 = bnr for n = 1, 2, ... , t - 1; its terms are determined by b1 and r.

N s%n 18. Discover and prove formulas similar to those of Problem 17 for the Lucas numbers Ln. 19. Use Example 4, in the text above, to prove the following properties of the Lucas numbers for n = 0, 1, 2, ... , and then prove them for all negative integers n. (a) Ln%4 ' 3Ln%2 & Ln. (b) Ln%6 ' 4Ln%3 % Ln. (c) Ln%8 ' 7Ln%4 & Ln. (d) Ln%10 ' 11Ln%5 % Ln. 20. State an analogue of Example 4 for the Fibonacci numbers instead of the Lucas numbers and use it to prove analogues of the formulas of Problem 19.

20. List the odd permutations of 1, 2, 3, 4. R 21. Let P be a permutation i, j, h, ... k of 1, 2, 3, ... , n. (a) Show that if i and j are interchanged, P changes from odd to even or from even to odd. (b) Show that if any two adjacent terms in P are interchanged, P changes from odd to even or from even to odd. (c) Show that the interchange of any two terms in P can be considered to be the result of an odd number of interchanges of adjacent terms. (d) Show that if any two terms in the permutation P are interchanged, P changes from odd to even or from even to odd.

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