104 3 Real Analysis
308.Leta= 4 k−1, wherekis an integer. Prove that for any positive integernthe
number
1 −(
n
2)
a+(
n
4)
a^2 −(
n
6)
a^3 +···is divisible by 2n−^1.309.LetAandEbe opposite vertices of a regular octagon. A frog starts jumping at
vertexA. From any vertex of the octagon exceptE, it may jump to either of the
two adjacent vertices. When it reaches vertexE, the frog stops and stays there.
Letanbe the number of distinct paths of exactlynjumps ending atE. Prove that
a 2 n− 1 =0 and
a 2 n=1
√
2
(xn−^1 −yn−^1 ), n= 1 , 2 , 3 ,...,wherex= 2 +√
2 andy= 2 −√
2.
310.Find all functionsf:N→Nsatisfying
f (f (f (n)))+ 6 f (n)= 3 f (f (n))+ 4 n+ 2001 , for alln∈N.- The sequence(xn)nis defined byx 1 =4,x 2 =19, and forn≥2,xn+ 1 =x
(^2) n
xn− 1 ,
the smallest integer greater than or equal to x
n^2
xn− 1. Prove thatxn−1 is always a
multiple of 3.
312.Consider the sequences given by
a 0 = 1 ,an+ 1 =
3 an+
√
5 a^2 n− 4
2,n≥ 1 ,
b 0 = 0 ,bn+ 1 =an−bn,n≥ 1.Prove that(an)^2 =b 2 n+ 1 for alln.3.1.3 Limits of Sequences......................................
There are three methods for determining the limit of a sequence. The first of them is
based on the following definition.
Cauchy’s definition.
(a)A sequence(xn)nconverges to a finite limitLif and only if for every> 0 there
existsn()such that for everyn > n(),|xn−L|<.