Question PLEASE REFER TO SOLUTION (4) 7. The Fibonacci sequence is given by Fi = 1. F2 = 1 and for n > 2. Fn+1 = F. +Fn-1. Thus {F}=1 = 1,1,2,3,5,8,13, 21, 34, 55, .... (Hint: See Ex. 1.5.6 for a similar example.) a) Let S denote the sum of the first n Fibonacci numbers, Sn:= I= Ft = Fi + F2 + ... + Fr. Calculate S. for n=1,2,..., 6. (Note, := means this is a definition.) b) Make a conjecture relating to Fn+2. c) Prove your conjecture in part (b) by induction. n 2 EXAMPLE 1.5.6. The Fibonacci sequence {Fr} = 1,1,2,3,5,8,13, 21, 34, 55, 89, 144, ..., is governed by the rule Fn+1 = Fn + Fr.-1 for n > 2. and the initial values Fi = F2 = 1. It is a sequence that arises many places in mathematics and in nature. For instance the ratios of successive Fibonacci numbers. Fr+i/Fr. approaches the Golden Ratio + = 1.61803... as n +00: 5 = 1.61764..., 3 = 1.61818..., and so on. Prove that (1.2) Fi + F3 +...+F2k-1 = F2k for any k eN. PROOF. Proof by induction on k. For k = 1 we have Fi = 1 = F2, so the true. Suppose that the statement (1.2) is true for a given k. Then for k+1 we have F1+F3 + ... + F2k-1 + F2k+1 = (F1+F3 + ... + F2k-1) + F2k+1 = F2x + F2k+1: by the induction hypothesis, = F2k+2 = F2(+1), by the defining property of the Fibonacci sequence. QED.
0L0DO4 The Asker · Advanced Mathematics
PLEASE REFER TO SOLUTION
Transcribed Image Text: (4) 7. The Fibonacci sequence is given by Fi = 1. F2 = 1 and for n > 2. Fn+1 = F. +Fn-1. Thus {F}=1 = 1,1,2,3,5,8,13, 21, 34, 55, .... (Hint: See Ex. 1.5.6 for a similar example.) a) Let S denote the sum of the first n Fibonacci numbers, Sn:= I= Ft = Fi + F2 + ... + Fr. Calculate S. for n=1,2,..., 6. (Note, := means this is a definition.) b) Make a conjecture relating to Fn+2. c) Prove your conjecture in part (b) by induction.
n 2 EXAMPLE 1.5.6. The Fibonacci sequence {Fr} = 1,1,2,3,5,8,13, 21, 34, 55, 89, 144, ..., is governed by the rule Fn+1 = Fn + Fr.-1 for n > 2. and the initial values Fi = F2 = 1. It is a sequence that arises many places in mathematics and in nature. For instance the ratios of successive Fibonacci numbers. Fr+i/Fr. approaches the Golden Ratio + = 1.61803... as n +00: 5 = 1.61764..., 3 = 1.61818..., and so on. Prove that (1.2) Fi + F3 +...+F2k-1 = F2k for any k eN. PROOF. Proof by induction on k. For k = 1 we have Fi = 1 = F2, so the true. Suppose that the statement (1.2) is true for a given k. Then for k+1 we have F1+F3 + ... + F2k-1 + F2k+1 = (F1+F3 + ... + F2k-1) + F2k+1 = F2x + F2k+1: by the induction hypothesis, = F2k+2 = F2(+1), by the defining property of the Fibonacci sequence. QED.
More Transcribed Image Text: (4) 7. The Fibonacci sequence is given by Fi = 1. F2 = 1 and for n > 2. Fn+1 = F. +Fn-1. Thus {F}=1 = 1,1,2,3,5,8,13, 21, 34, 55, .... (Hint: See Ex. 1.5.6 for a similar example.) a) Let S denote the sum of the first n Fibonacci numbers, Sn:= I= Ft = Fi + F2 + ... + Fr. Calculate S. for n=1,2,..., 6. (Note, := means this is a definition.) b) Make a conjecture relating to Fn+2. c) Prove your conjecture in part (b) by induction.
n 2 EXAMPLE 1.5.6. The Fibonacci sequence {Fr} = 1,1,2,3,5,8,13, 21, 34, 55, 89, 144, ..., is governed by the rule Fn+1 = Fn + Fr.-1 for n > 2. and the initial values Fi = F2 = 1. It is a sequence that arises many places in mathematics and in nature. For instance the ratios of successive Fibonacci numbers. Fr+i/Fr. approaches the Golden Ratio + = 1.61803... as n +00: 5 = 1.61764..., 3 = 1.61818..., and so on. Prove that (1.2) Fi + F3 +...+F2k-1 = F2k for any k eN. PROOF. Proof by induction on k. For k = 1 we have Fi = 1 = F2, so the true. Suppose that the statement (1.2) is true for a given k. Then for k+1 we have F1+F3 + ... + F2k-1 + F2k+1 = (F1+F3 + ... + F2k-1) + F2k+1 = F2x + F2k+1: by the induction hypothesis, = F2k+2 = F2(+1), by the defining property of the Fibonacci sequence. QED.
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Given the Fibonacci series, the sum is given by  a) S_(1)=F_(1)=1,[F_(3)=1:} S_(2)=S_(1)+F_(2)=2,[F_(4)=3:} S_(3)=S_(2)+F_(3)=4,[F_(5)=5:} S_(4)=S_(3)+F_(4)=7,[F_(6)=8:} S_(5)=S_(4)+F_(5)=12,[F_(7)=13:} S_(6)=S_(5)+F_(6)=20,[F_(8)=21:} b) From the above values, we can conjecture S_(n)=F_(n+2)-1 c) For, k = 1  S_(1)=1,F_(1+2)=F_(3 ... See the full answer