98 3 Real Analysis
3.1 Sequences and Series...........................................
3.1.1 Search for a Pattern.......................................
In this section we train guessing. In each problem you should try particular cases until
you guess either the general term of a sequence, a relation that the terms satisfy, or an
appropriate construction. The idea to write such a section came to us when we saw the
following Putnam problem.
Example.Consider the sequence(un)ndefined byu 0 =u 1 =u 2 =1, and
det(
un+ 3 un+ 2
un+ 1 un)
=n!,n≥ 0.Prove thatunis an integer for alln.
Solution.The recurrence relation of the sequence is
un+ 3 =
un+ 2 un+ 1
un+
n!
un.
Examining some terms:
u 3 =1 · 1
1
+
1
1
= 2 ,
u 4 =2 · 1
1
+
1
1
= 3 ,
u 5 =3 · 2
1
+
2
1
= 4 · 2 ,
u 6 =4 · 2 · 3
2
+
3 · 2
2
= 4 · 3 + 1 · 3 = 5 · 3 ,
u 7 =5 · 3 · 4 · 2
3
+
4 · 3 · 2
3
= 5 · 4 · 2 + 4 · 2 = 6 · 4 · 2 ,
u 8 =6 · 4 · 2 · 5 · 3
4 · 2
+
5 · 4 · 3 · 2
4 · 2
= 6 · 5 · 3 + 5 · 3 = 7 · 5 · 3.
we conjecture that
un=(n− 1 )(n− 3 )(n− 5 )···.This formula can be proved by induction. Assuming the formula true forun,un+ 1 , and
un+ 2 , we obtain
un+ 3 =un+ 2 un+ 1 +n!
un=
(n+ 1 )(n− 1 )(n− 3 )···n(n− 2 )(n− 4 )···+n!
(n− 1 )(n− 3 )(n− 5 )···