27. Find numbers s and t such that n³ = n(n - 1)(n - 2) + sn(n - 1) + tn holds for n = 1 and n = 2.

28. Find numbers a and b such that  for all integers n.

29. Find numbers r, s, and t such that n⁴ = n(n - 1)(n - 2)(n - 3) + rn(n - 1)(n - 2) + sn(n - 1) + tn for n = 1, 2, and 3. Using these values of r, s, and t, show that

for all integers n.

30. Find numbers a, b, c, and d such that

31. Express  as a polynomial in n.

32. Express  as a polynomial in n.

33. We define a sequence S₀, S₁, S₂, ... as follows: When n is an even integer 2t, let

When n is an odd integer 2t + 1, let 

Prove that S_2t + S_2t+1 = S_2t+2 and S_2t+1 + S_2t+2 = S_2t+3 for t = 0, 1, 2, ... .
 
 

Monday, June 8, 1998