d) Consider the sequence Dn, where Dn describes the number of ways to paint the

Question

d) Consider the sequence Dn, where Dn describes the number of ways to paint the floors on an n-story building where each floor is painted yellow or blue and no two adjacent floors can be blue (although adjacent floors can be yellow). For example, a 3-story building can be painted in the following ways (each column represents a different way): MODULE CODE& TITLE: MB5014 Problem Solving and Mathematical Modelling 3rd Floor Y Y Y B B 2nd Floor Y Y BY Y 1st FloorY B Y B Y (i) Determine the values of the initial conditions D and D2 from the above information (ii) Write down the second order iterative equation which models the situation in the question. Explain why this is the correct equation

Explanation / Answer

i) D1=2 since there are two options Y or B.

D2=3 since there are 3 options {YY},{YB},{BY}

ii) In Dn the nth floor can be coloured with Y or B.If it is coloured with Y the remaining n-1 floors can be coloured in Dn-1 ways . If nth floor is coloured B then n-1th floor must be coloured Y. The remaining n-2 floors can be coloured in Dn-2 ways . Therefore the n floors can be coloured in Dn-1 +Dn-2 ways since both the cases are mutually exclusive and exhaustive.

hence Dn =  Dn-1 +Dn-2

This is the required second order iterative equation.

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