Question

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 $0 .$ Which of the following is correct?

• A. $a_{17}=a_{16}+a_{15}$
• B. $c_{17} \neq c_{16}+c_{15}$
• C. $b_{17} \neq b_{16}+c_{16}$
• D. $a_{17}=c_{17}+b_{16}$

Answer 1

Answer: (A)

$a_{17}=a_{16}+a_{15}$

Answer 2

As $a_{n}=a_{n-1}+a_{n-2}$ for $n=17$ $\Rightarrow a_{17}=a_{16}+a_{15}$