By Abraham P Hillman
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N 3 % an 2 % bn % c for all integers n. n 4 ' n 4 % pn 3 % qn 2 % rn % s for all integers n. 5. Prove that n 3 ' 0 for n ' 0, 1, 2. n k 6. Given that k is a positive integer, prove that 7. Find &1 r for r ' 0, 1, 2, 3, 4, and 5. 8. Find &2 r for r ' 0, 1, 2, 3, 4, and 5. 9. Prove that &3 r ' (&1)r r%2 2 ' 0 for n ' 0, 1, ... , k & 1. for r ' 0, 1, 2, ... 47 &4 r 10. Prove that r%3 3 ' (&1)r for r ' 0, 1, 2, ... 11. Let m be a positive integer and r a non-negative integer. Express n k binomial coefficient &m r in terms of a with 0 # k # n.
Prove the following for all integers m and n: (a) Lm%n%1 ' Fm%1Ln%1 % FmLn. (b) Fm%n%1 ' Fm%1Fn%1 & FmFn. 37 24. Prove that (Fn+1)2 + (Fn)2 = F2n+1 for all integers n. 25. Let a and b be the roots of the quadratic equation x2 - x - 1 = 0. Prove that: (a) Fn ' an & bn . a &b (b) Ln ' a n % b n. (c) FnLn ' F2n. (d) a n ' aFn % Fn&1 and b n ' bF n % Fn&1. 26. The sequence 0, 1, ½, 3/4, 5/8, 11/16, ... is defined by u0 ' 0, u1 ' 1, u2 ' u1 % u0 un%1 % un , ÿ, un%2 ' 2 2 Discover and prove a compact formula for un as a function of n.
Let us consider the set of four objects a, b, c, d. They can be arranged in the following ways: abcd bacd cabd dabc abdc badc cadb dacb acbd bcad cbad dbac acdb bcda cbda dbca adbc bdac cdab dcab adcb bdca cdba dcba 53 Rather than write them all out, if we are only interested in the number of arrangements, we may think of the problem thus: We have four spaces to fill. If we put, for example, the b in the first, we have only the a, c, and d to choose from in filling the remaining three. And if we put the d in the second, we have only a and c for the remaining; and so forth.