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 $b_n$ denote the number such n-digit integers ending with digit 1 and $c_n$ denote the number of such n-digit integers ending with digit 0. The value $b_6$ is _______.
Answer & explanation
Clearly,
$a_n = b_n + c_n, b_n =a_{n-1}$ and $c_n = a_{n-2}$
$∴a_n = a_{n-1}+ a_{n-2}$
Also, $a_1 =1, a_2 = 2$
$∴a_3 =2+1=3, a_4=a_3+ a_2 = 3+2=5$,
$a_5=a_4+a_3 =5+3=8$
$∴b_6=a_5=8$