Question

$n$ digit numbers are formed using the digits 1, 2,3,4 each of which begin with 1 such that no two consecutive digits are same. If $a_n(n\ge 2)$ be such $n$-digit numbers which ends in 1 while $b_n$ be the ones which ends in a fixed number other than 1 , then
The value of $a_7$ equals

• A. 180
• B. 183
• C. 190
• D. 197

Answer 1

Answer: (B)

$\text{ Clearly, }{a}_n=3b_{n-1}\Rightarrow a_{n+1}=3b_n$
$b_n=a_{n-1}+2b_{n-1}\Rightarrow a_{n+1}=a_{n-1}+\frac{2}{3}{a}_n$
$\Rightarrow a_{n+1}=3a_{n-1}+2a_n$
Clearly $a_2=0,a_3=3$
$\Rightarrow a_4=3a_2+2a_3=6$
$a_5=3a_3+2a_4=9+12=21$
$a_6=3a_4+2a_5=18+42=60$
$a_7=3a_5+2a_6=63+120=183$