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)

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