Using index notation, show that the trace is cyclic, i.e. that for any number m of N × N matrices M(1), M(2),..., M(m), the following holds true: Tr ( M (1) M (²) ... M(m)) = Tr (M(m) M(¹) M (²) ... M(m−1)) (1)

Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.

Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection with this result, see the discussion of counterexamples in the Appendix.) 1+2n2n for all integers n3

The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is defined recursively by f1=1,f2=1,fn+2=fn+1+fn for n=1,2,3,... a. Prove f1+f2+...+fn=fn+21 for all positive integers n. b. Use complete induction to prove that fn2n for all positive integers n. c. Use complete induction to prove that fn is given by the explicit formula fn=(1+5)n(15)n2n5 (This equation is known as Binet's formula, named after the 19th-century French mathematician Jacques Binet.)