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Question Number 21913 by Tinkutara last updated on 06/Oct/17

$$\mathrm{Let}{a}_n\mathrm{denote}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{all}n-\mathrm{digit}$$$$\mathrm{positive}\mathrm{integers}\mathrm{formed}\mathrm{by}\mathrm{the}\mathrm{digits}$$$$0,1\mathrm{or}\mathrm{both}\mathrm{such}\mathrm{that}\mathrm{no}\mathrm{consecutive}$$$$\mathrm{digits}\mathrm{in}\mathrm{them}\mathrm{are}0.\mathrm{Let}{b}_n=\mathrm{the}$$$$\mathrm{number}\mathrm{of}\mathrm{such}n-\mathrm{digit}\mathrm{integers}\mathrm{ending}$$$$\mathrm{with}\mathrm{digit}1\mathrm{and}{c}_n=\mathrm{the}\mathrm{number}\mathrm{of}$$$$\mathrm{such}n-\mathrm{digit}\mathrm{integers}\mathrm{ending}\mathrm{with}$$$$\mathrm{digit}0.$$$$1.\mathrm{Which}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{is}\mathrm{correct}?$$$$\left(1\right)a_{17}=a_{16}+a_{15}$$$$\left(2\right)c_{17}\ne c_{16}+c_{15}$$$$\left(3\right)b_{17}\ne b_{16}+c_{16}$$$$\left(4\right)a_{17}=c_{17}+b_{16}$$$$2.\mathrm{The}\mathrm{value}\mathrm{of}{b}_6\mathrm{is}$$

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