Math Answers

. Show that the sequence {fn} defined explicitly by fn= 2(-4)n +3 is a

solution of the recurrence relation fn = -3 fn-1+4 fn-2.

3. Suppose that the discrete function f is defined recursively by:

f(0) =2 and

f(n+1) = 2f(n) +3

Then find f(1), f(2), f(3), f(4) and f(5)

4. Find a recursive definition for the following sequence

a. an = {2,4,8,16,32, …}, where n 1,

b. an ={3,6,9,12,15, … ) , where n 1,

5. Give a recursive definition of the sequence {fn} ,  if

(a) fn =5n (b) fn = 2n+1

(c) fn+1 = 10n (d) fn = 5.

6. Find the characteristic equation of each of the following recurrence

relations

a. fn = 3fn-1 -2fn-2, n > 2 with initial conditions: f1=3 and f2 = 7

b. yn= 6yn-1- 9yn-2, n>1, y0= 5 and y1= 3

7. Solve the following recurrence relations.

(a) an =5an-1 +6an-2, n  2 with a0 =1, a1= 3.

(b) 2an+2 –11an+1+ 5an = 0, n  0 with a0= 2, a1= -8.

(c) 3an+1= 2an+ an-1, n  1, a0 =7, a1 = 3

(d) an – 6an-1+ 9an-2 = 0, n  2, a0 =5, a1 =12.

George Drug Products Ltd (GDPL) is faced with several possible investment projects. For each, the total cash outflows required will occur in the initial period. The cash outflows expected net present values and standard deviations are as follows:

Project Cost Sh. ‘000’ Net present value Standard deviations

A 10,000   1,000         2,000

B  5,000             1,000                   3,000

      C               20,000             2,500                  1,000

D                1,000               500                  1,000

E               50,000              7,500                  7,500

All projects have been discounted at a risk-free rate of 8% and it is assumed that the distribution of their possible net present values are normal.

(a) Construct a risk profile for each of these projects n terms of the profitability index     (8 marks)

(b) Ignoring size problems do you find some projects clearly dominated by others? (7 marks)