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≥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 a₇ equals

a) 180

b) 183 

c) 190 

d) 197

Answer 1

Correct option is (b) 183

Clearly, a_n = 3b_n–1

⇒ a_n+1 = 3b_n

b_n = a_n–1 + 2b_n–1

⇒ a_n+1 = a_n–1 + \(\frac 23\)a_n

⇒ a_n+1 = 3a_n–1 + 2a_n

Clearly a₂ = 0, a₃ = 3

⇒ a₄ = 3a₂ + 2a₃ = 6

a₅ = 3a₃ + 2a₄ = 9 + 12 = 21.

a₆ = 3a₄ + 2a₅ = 18 + 42 = 60.

a₇ = 3a₅ + 2a₆ = 63+ 120  = 183.