Let p(n, k) be the number of ways to write the integer n as a sum of exactly k positive integers. A theorem later in the text tells us that: p(n, k) = p(n – 1, k – 1) + p(n – k, k) Also, p(n, k) O when k >n and p(n, 0) = 0 and p(n, n) = 1. Write a recursive algorithm to find p(n, k).

Find the general solution of the recursive relation an an = a10 (3)" + α20 (-3)" an = (a10+ a11 n) (-3)" None of these. an = α10 (6)" + α20 (-9)" an = (α10+ α11 n) (3)" = 6an-19an-2, n ≥ 2.

Use Gauss’s trick to determine H(n) = 1 + 5 + 9 + 13 + … + (4n – 7) + (4n – 3) + (4n + 1) I'm stuggling to know how many to multiply the sum by 2H(n) = (4n+2) (?) Is it n or n+1 or 4n+1

Let x≥2, conjecture and prove an expression with no summation signs and no recursviely calculated values for S(x, 2), and S(x, x −2). Suppose x≥1, conjecture and prove an expression with no summation signs and no recursviely calculated values for c(x, x-1) note: S(x, x-1) = (x choose 2) this is a combinatorics math problem

show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n (1/2)3n+2))/(3n+2) )

19. Let f(n) be defined recursively by f(0) = 3, and f(n + 1) = 3f(n)/3, for n = 0, 1, 2, . . . Then, f(10) = _____________.