1. Tardigrade
  2. Question
  3. Mathematics
  4. 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 an(n ≥ 2) be such n-digit numbers which ends in 1 while bn be the ones which ends in a fixed number other than 1 , then The value of a 7 equals

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. $n$ 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 $a_n(n \geq 2)$ 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

Permutations and Combinations

Solution:

$\text { Clearly, } a_n=3 b_{n-1} \Rightarrow a_{n+1}=3 b_n$
$b _{ n }= a _{ n -1}+2 b _{ n -1} \Rightarrow a _{ n +1}= a _{ n -1}+\frac{2}{3} a _{ n } $
$\Rightarrow a _{ n +1}=3 a _{ n -1}+2 a _{ n }$
Clearly $a _2=0, a _3=3$
$\Rightarrow a_4=3 a_2+2 a_3=6 $
$a_5=3 a_3+2 a_4=9+12=21 $
$a_6=3 a_4+2 a_5=18+42=60 $
$a_7=3 a_5+2 a_6=63+120=183$