Let an 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 bn= the number of such n - digit integers ending with digit 1 and cn= the number of such n-digit integers ending with digit 0 . Which of the following is correct?

Question

Let an 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 bn= the number of such n - digit integers ending with digit 1 and cn= the number of such n-digit integers ending with digit 0 . Which of the following is correct? (A) a17=a16+a15 (B) c17 ≠ c16+c15 (C) b17 ≠ b16+c16 (D) a17=c17+b16

Tardigrade Admin · Accepted Answer

as an=an-1+an-2, for n =17 ⇒ a17=a16+a15 .

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

$c_{17} \neq = c_{16} + c_{15}$

$b_{17} \neq = b_{16} + c_{16}$

$a_{17} = c_{17} + b_{16}$

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

a17​=a16​+a15​

c17​=c16​+c15​

b17​=b16​+c16​

a17​=c17​+b16​

as an​=an−1​+an−2​, for n=17⇒a17​=a16​+a15​.

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

$c_{17} \neq = c_{16} + c_{15}$

$b_{17} \neq = b_{16} + c_{16}$

$a_{17} = c_{17} + b_{16}$

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