1. Prove by induction that (-1) + 3 + 7 + ... + (4n-5) = 2n

2- 3n for all natural numbers n. The k-th term in the sum is 2k-5.

2. (a) Prove that for every real number x in the interval [-2, 4], |x + 2| is

greater than or equal to 4 or |x - 4| is greater than or equal to 2.

(b) Prove that n^3- n + 4 is an even number for every integer n.

3. Suppose that a is congruent to 4 and b is congruent to 6 modulo 7. Find

the residue of (a) a - b mod 7;

(b) ab mod 7.

Give a proof.

4. Show that there is no rational number r satisfying r^3= 5.