3.1 Sequences and Series 99=
(n+ 1 )·n!+n!
(n− 1 )(n− 3 )(n− 5 )···=
(n+ 2 )n!
(n− 1 )(n− 3 )(n− 5 )···
=(n+ 2 )n(n− 2 )(n− 4 )···.This completes the induction, and the problem is solved.
297.Find a formula for the general term of the sequence
1 , 2 , 2 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 5 , 5 , 5 , 5 , 5 ,....298.Find a formula in compact form for the general term of the sequence defined re-
cursively byx 1 =1,xn=xn− 1 +nifnis odd, andxn=xn− 1 +n−1ifnis
even.
299.Define the sequence(an)n≥ 0 bya 0 =0,a 1 =1,a 2 =2,a 3 =6, and
an+ 4 = 2 an+ 3 +an+ 2 − 2 an+ 1 −an, forn≥ 0.Prove thatndividesanfor alln≥1.300.The sequencea 0 ,a 1 ,a 2 ,...satisfies
am+n+am−n=1
2
(a 2 m+a 2 n),for all nonnegative integersmandnwithm≥n.Ifa 1 =1, determinean.301.Consider the sequences(an)n,(bn)n, defined by
a 0 = 0 ,a 1 = 2 ,an+ 1 = 4 an+an− 1 ,n≥ 0 ,
b 0 = 0 ,b 1 = 1 ,bn+ 1 =an−bn+bn− 1 ,n≥ 0.Prove that(an)^3 =b 3 nfor alln.302.A sequenceunis defined by
u 0 = 2 ,u 1 =5
2
,un+ 1 =un(u^2 n− 1 − 2 )−u 1 , forn≥ 1.Prove that for all positive integersn,un= 2 (^2
n−(− 1 )n)/ 3
,where·denotes the greatest integer function.303.Consider the sequences(an)nand(bn)ndefined bya 1 =3,b 1 =100,an+ 1 = 3 an,
bn+ 1 = 100 bn. Find the smallest numbermfor whichbm>a 100.