3.1 Sequences and Series 103vn+ 1 =un+vn.Combining the two, we obtain the (vector-valued) recurrence relation
(
un+ 1
vn+ 1
)
=
(
32
11
)(
un
vn)
.
The characteristic equation, of the coefficient matrix but also of the sequencesunand
vn,is
∣
∣∣
∣
λ− 3 − 2
− 1 λ− 1∣
∣∣
∣=λ(^2) − 4 λ+ 1 = 0.
Its roots areλ 1 , 2 = 2 ±
√
- We compute easilyu 1 =3 andv 1 =1, sou 2 = 3 · 3 + 2 · 1 =11.
The desired general-term formula is then
un=1
2
√
3
((√
3 + 1
)(
2 +
√
3
)m
+(√
3 − 1
)(
2 −
√
3
)m).
Figure 18Below are listed more problems of this kind.304.Letp(x)=x^2 − 3 x+2. Show that for any positive integernthere exist unique
numbersanandbnsuch that the polynomialqn(x)=xn−anx−bnis divisible
byp(x).
305.Find the general term of the sequence given byx 0 =3,x 1 =4, and
(n+ 1 )(n+ 2 )xn= 4 (n+ 1 )(n+ 3 )xn− 1 − 4 (n+ 2 )(n+ 3 )xn− 2 ,n≥ 2.306.Let(xn)n≥ 0 be defined by the recurrence relationxn+ 1 =axn+bxn− 1 , withx 0 =0.
Show that the expressionxn^2 −xn− 1 xn+ 1 depends only onbandx 1 , but not ona.
307.Define the sequence(an)nrecursively bya 1 =1 and
an+ 1 =1 + 4 an+√
1 + 24 an
16
, forn≥ 1.Find an explicit formula foranin terms ofn.