Advanced High-School Mathematics

84 CHAPTER 2 Discrete Mathematics

1

x 1

+

1

x 2

+···+

1

xn

+

1

xn+1

≥

n^2

1 −xn+1

+

1

xn+1

.

Next, you need to argue that because 0< xn+1<1,

n^2

1 −xn+1

+

1

xn+1

≥(n+ 1)^2 ;

this is not too difficult. Incidently, when does equality occur in

the above inequality?)

7. Prove that for all integersn≥1, 2

∑n

j=1

sinxcos^2 j−^1 x = sin 2nx.

8. Prove that for all integersn≥0, sinx

∏n

j=0

cos 2jx =

sin

Ä

2 n+1x

ä

2 n+1

.

9. Prove that for all integersn≥0, that

∑n

j=1

sin(2j−1)x =

1 −cos 2nx

2 sinx

.

10. (This is a bit harder.) Prove the partial fraction decomposition

1

x(x+ 1)(x+ 2)···(x+n)

=

1

n!

∑n

k=0

(−1)k

Ñ

n

k

é

1

x+k

,

wherenis a non-negative integer.

11.^15 We shall use mathematical induction to prove that all positive
integers are equal. LetP(n) be the proposition

P(n) :

“If the maximum of two positive

integers is n then the integers are

equal.”

(^15) Due to T.I. Ramsamujh,The Mathematical Gazette, Vol. 72, No. 460 (Jun., 1988), p.
113.