31. Verify the factorization 1 - x⁷ = (1 - x)(1 + x + x² + x³ + x⁴ + x⁵ + x⁶) and use it with x = 1/2 to find a compact expression for

32. Use the factorization 1 + x⁹⁹ = (1 + x)(1 - x + x² - x³ + x⁴ - ... + x⁹⁸) to find compact expressions for the following sums:

(a) 1 - 5^-1 + 5^-2 - 5^-3 + ... - 5^-97 + 5^-98.

(b) a - ar + ar² - ar³ + ... - ar⁹⁷ + ar⁹⁸.

33. Let a₁, a₂, a₃, ..., a_3m be an arithmetic progression, and for n = 1, 2, ..., 3m let A_n be the arithmetic mean of its first n terms. Show that A_2m is the arithmetic mean of the two numbers A_m and A_3m.

34. Let g₁, g₂, ..., g_3m be a geometric progression of positive terms. Let A, B, and C be the geometric means of the first m terms, the first 2m terms, and all 3m terms, respectively. Show that B² = AC.

35. Let S be the set consisting of those of the integers 0, 1, 2, ..., 30 which are divisible exactly by 3 or 5 (or both), and let T consist of those divisible by neither 3 nor 5.

(a) Write out the sequence of numbers in S in their natural order.

(b) In the sequence of Part (a), what is the arithmetic mean of terms equally spaced from the two ends of the sequence?

(c) What is the arithmetic mean of all the numbers in T?

(d) Find the sum of the numbers in T.

36. Find the sum 4 + 5 + 6 + 8 + 10 + 12 + 15 + ... + 60,000 of all the positive integers not exceeding 60,000 which are integral multiples of at least one of 4, 5, and 6.

37. Let u₁, u₂, ..., u_t satisfy u_n+2 = 2u_n+1 - u_n for n =1, 2, ..., t - 2. Show that the t terms are in arithmetic progression.
 

Sunday, March 15, 1998