Advanced High-School Mathematics

SECTION 2.1 Elementary Number Theory 83

1. Prove the following:
   (i) 1 + 3 + 5 +···+ (2n−1) =n^2 (n= 1, 2 ,...)
   (ii) 1^3 + 2^3 + 3^3 +···+n^3 =^14 n^2 (n+ 1)^2 (n= 1, 2 ,...)

(iii)

1

1 · 3

+

1

3 · 5

+···

1

(2n−1)(2n+ 1)

=

n

2 n+ 1

(n= 1, 2 ,...).

(Do you really need mathematical induction? Try partial frac-

tions!)

(iv) 1^2 +

(

1

2

) 2

+

(

1

3

) 2

+···+

(

1

n

) 2

< 2 −

1

n

(n= 2, 3 ,...)

2. As in Exercise 6 on page 78 we define, for any positive integern,

H(n) = 1 +

1

2

+

1

3

+···+

1

n

.

Show that for any integerm≥0, thatH(2m)≥

m+ 2

2

.

3. Letnbe a positive integer.

(a) Prove that ifkis an integer with 0≤k≤n,

Ñ

n

k

é

=

Ñ

n− 1

k

é

+

Ñ

n− 1

k− 1

é

. (This doesn’t require induction.)

(b) Prove that ifSis a set withnelements, and if 0≤k≤n, then

there are

Än

k

ä

subsets ofSwithkelements. (Use induction.)

4. Prove that for alln≥ 1 ., 13 + 2^3 +···n^3 = (1 + 2 + 3 +···+n)^2.


5. Prove that for alln≥ 1 ,and for allx≥0, that (1 +x)n>1 +nx.
   (Is induction really needed?)


6. Prove the classical inequality

1

x 1

+

1

x 2

+···+

1

xn

≥n^2

wheneverx 1 , x 2 , ...xn>0 andx 1 +x 2 +···xn= 1. (Hint: using

induction, note first that you can arrive at the inequality