MATH 295 PROBLEMS 1

IGOR KRIZ

1. Find all real numbers x for which (x − 1)(x − 3) > 0.

2. Can you find a reason why the following formula holds:

1 + a + ... + a^{n} = \frac{1 − a^{n+1}}{1 − a}?

(a is a real number, n is a positive integer)

3. Recall that |a + b| ≤ |a| + |b|. Using this fact, show that

||a| − |b|| ≤ |a + b| .

(| a | is a if a ≥ 0 and −a else for a real number a.)

4. What is wrong with the following “proof”? Let x = y. Then

x^{2} = xy x^{2} − y^{2} = xy − y^{2} (x + y)(x − y) = y(x − y) x + y = y 2y = y 2 = 1.