38. Find a compact expression for the sum v₁ + v₂ + ... + v_t in terms of v₁ and v₂, given that v_n+2 = (v_n+1)²/v_n for n = 1, 2, ... ,t - 2.

39. Let a_n = 2ⁿ be the nth term of the geometric progression 2, 2², 2³, ... , 2^t. Show that a_n+2 - 5a_n+1 + 6a_n = 0 for n = 1, 2, ... , t - 2.

40. For what values of r does the sequence b_n = rⁿ satisfy b_n+2 - 5b_n+1 + 6b_n = 0 for all n?

41. Let a be one of the roots of x² - x - 1 = 0. Let the sequence c₀, c₁, c₂, ... be the geometric progression 1, a, a², ... . Show that:

(a) c_n+2 = c_n+1 + c_n.

(b) c₂ = a² = a + 1.

(c) c₃ = a³ = 2a + 1.

(d) c₄ = 3a + 2.

(e) c₅ = 5a + 3.

(f) c₆ = 8a + 5.

42. For the sequence c₀, c₁, ... of the previous problem, express c₁₂ in the form aF_u + F_v, where F_u and F_v are Fibonacci numbers, and conjecture a similar expression for c_m.

^*43. In the sequence 1/5, 3/5, 4/5, 9/10, 19/20, 39/40, ... each succeeding term is the average of the previous term and 1. Thus:

(a) Show that the twenty-first term is 

(b) Express the nth term similarly.

(c) Sum the first five hundred terms.

^*44. In the sequence 1, 2, 3, 6, 7, 14, 15, 30, 31, ... a term in an even numbered position is double the previous term, and a term in an odd numbered position (after the first term) is one more than the previous term.

(a) What is the millionth term of this sequence?

(b) Express the sum of the first million terms compactly.
 
 

Sunday, March 15, 1998