17. (a) Given that  show that 

(b) Given that  show that 

(c) Given that  and a < c, show that 

R 18. Given that  and a₁ < a_n, show that
a_n > (a₁ + a₂ + ... + a_n-1)/(n - 1).

19. Find integers a, b, and c such that 0 < a < b < c, a + b > c, and c is as small as possible.

20. Let m and n be positive integers, and let 1, m, and n be the lengths of the sides of a triangle. Show that m = n.

21. Given that x > 0 and y > 0, show that (x + y)ⁿ > x^n-1(x + ny) for all integers 

22. Given that  prove by mathematical induction that  for all positive integers n.

23. Prove that  for all positive real numbers x.

24. Prove that  for all positive integers n.

25. Prove that  for all positive integers n.

26. Use the fact that 1<b and x<y imply b^x<b^y to prove the inequalities 

for the sequence a₁, a₂, ... defined by

27. Use the fact that 0 < b < 1 and x < y imply b^x > b^y to prove the inequalities

for the sequence u₁, u₂, ... defined by

Monday, June 22, 1998