Let a_ n denote the number of all n -digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let b_ n = the number of such n -digit integers ending with digit 1 and c_ n = the number of such n -digit integers ending with digit 0 . Question: The value of b₆ is

Step-by-step solution

a_ n = number of all n digit +ve integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . and b_ n = number of such n digit integers ending with 1 c_ n = number of such n digit integers ending with 0 Clearly, a_ n =b_ n +c_ n ( a_ n . can end with 0 or 1) Also b_ n =a_ n-1 and c_ n =a_ n-2 [ if last digit is 0 , second last has to be 1] We get a_ n =a_ n-1 +a_ n-2 , n 3 Also a₁=1, a₂=2 , Now by this recurring formula, we get aligned a₃ &=a₂+a₁=3 a₄ &=a₃+a₂=3+2=5 a₅ &=a₄+a₃=5+3=8 Als