Question

can someone help me with this two questions please

thank you

4. Find a recurrence relation (with initial conditions) for an, the number of ternary sequences of length n that do not contain three consecutive digits that are the same. That is, the patterns '000','111', 222 must not appear anywhere in the sequence. So, 0011012 is acceptable, but 000022 and 1000112 are not. 5. Elsa is making trains out of colored train cars: the red cars are 2 inches long, the blue cars are also 2 inches long, and the purple cars are 3 inches long. Find a recurrence relation (with initial conditions) for to the number of trains of length n inches that can be made.

Answer 1

Page Nol = | The number of ternary sequences of length. that do not contain three consective digits Seh Each term of I with term = 3 2 زار و ie find a recurrence relation for an such sequence is either oor 1082. :, a, = Number of sequences of length o , co 2 sequences are a = number of sequences of length a x, y cosa, y =2) A₂ = 3x3 = 2₂=9 az - All sequences of fesom x,y,z (4,36 0,1,23 except ODIH, 222 =) 93 = 3X3X3- 3 = A=24 x Number of sequences of length in with first termo y = Number of an's with first term 1 Number of aus with first terma an = x + y + 3 If in all sequences in a we replace obyl, I by 2 2 by All these will convert sequences to sequences starting with x=y Let n74 Let 3= and 2

Page Noz Also is in all in all sequences in a we replace oby a aby, and I by o; All these sequence will convert torsequences with first terma x=3 x=ytz where x = Number of sequence in an with first termo are of types = 3x To count x Those sequences 1 an2 Theis number = 3 Their number = an-2 3 1 Their number any 02 m - Their number ant 3 I m 3 Rectangle has (-2) terms starting with I 2. Their number = an-2 is identical Rectangle has (1) terms, starting with Their number ant same is in ) an =3x=3 [ 92 M2+ anz =) an 20m-2 tan-1) § 33 a,=3, 22=9, an=2 (anx+m2) for all n73 + any ie 9,=3, A2=9, 93=2/3+9)=24, au= 2/9+24) = 63 as = 2(63+24)=174