Recurrence relation

1. Pauline takes a loan of S dollars at an interest rate of r percent per month, compounded monthly. She plans to repay the loan in T equal monthly installments of P dollars each. a) Let a(subscript n) denote the amount Pauline owes on the loan after n months. Write a recurrence relation for a (subscript n). b) Solve the recurrence relation that you obtained in a), above. c) Use your answer from b) to obtain a formula for the monthly payment, P.

Combinatorial mathematics

6. Recall that R^3={(x,y,z):x,y,z(subset of R)}. Let G(V,E) be a directed graph, in which V= {(x,y,z)-(subset of R^3) :x,y,z(subset of R),-10<=x,y,z<=10}. Suppose that for any vertex, v=(x,y,z)--[subset of V], the only edges originating at v are the ones joining v to (x+1,y,z),(x,y+1,z),(x,y,z+1) . i.e. any path that originates at v , must begin by moving one unit horizontally to (x+1,y,z) or one unit vertically to (x,y+1,z) or one unit across to (x,y,z +1) . a) How many distinct paths exist between(-1,2,0) and (1,3,7) ? b) How many distinct paths exist between(1,0,5) and (8,1,7) ?

To find the number of monomials of length n, to write a generating function.

Note that the generating function has to be in terms of powers of x. Example: the number of ways to select r balls from a pile of three green, three white, three blue, and three gold balls is the generating function--->(x^0+x^1+x^2+x^3)^4 Here's the problem: 4. In noncommutative algebra, the term monomial refers to any arrangement of a sequence of variables from a set. For example, in a noncommutative algebraic structure on a set of four variables, {x,y,z,w} , examples of monomials of length 3 are xxx,xyx,xxy,zwy,wzx,... a) Write a generating function for the number of monomials of length, n, in a noncommutative algebraic structure on a set of four variables. b) Find the number ... click for more

Prove recurrence by induction

Prove using induction that the recurrence T(n) = T(p*n) + t(q*n) + c*n for n > 1, T(1) = c / (1-p-q), for positive constants c,p,q such that p + q < 1 has the solution: (c*n) / (1-p-q)

Graph Problem with Depth First Search and Breadth First Search

Let G = (V, E) be a connected and undirected graph, and u is a chosen vertex in V. Suppose starting from u, exactly the same tree T is obtained using either breadth first search or depth first search. Prove that G = T, where T is the BFS or DFS tree.