- Submitted By: Nix24
- Date Submitted: 06/16/2014 12:22 PM
- Category: Miscellaneous
- Words: 1494
- Page: 6
- Views: 3

1. [9 marks]

(a) Find, showing all working, a formula for the n-th term tn of the sequence (tn) defined by

t1 = 8; tn = 5tn1/2, n 2.

Solution (4 marks):

t1 = 8, t2 =

5

2

t1 =

5

2

8

t3 =

5

2

t2 =

5

2

2

8

t4 =

5

2

t3 =

5

2

3

8 and so on . . .

In general, tn = 8

5

2

n1 for n 1 (geometric progression).

(b) Find, showing all working, a recursive definition of the sequence with general term

tn = 2 (n + 1)! 5n, n 1.

Solution (5 marks):

We have t1 = 2 2! 51 = 2 2 5 = 20, and looking at the ratio of successive terms:

tn/tn1 =

6 2 (n + 1)! 5n

6 2 (n 1 + 1)! 5n1

=

(n + 1)! 5n

n! 5n1

=

(n + 1)n! 5 5n1

n!5n1

= (n + 1)5

Hence, tn = 5(n + 1) tn1 for n 2. Therefore, a recursive definition of the sequence (tn) is

t1 = 20; tn = 5(n + 1) tn1, n 2.

1

2. [15 marks] On the first day (day 1) after grape harvesting is completed, a grape grower has 8000 kg

of grapes in storage. At the end of day n, for n = 1, 2, . . . , the grape grower sells 250n/(n + 1) kg of

their stored grapes at the local market at the price of $1.50 per kg. During each day the stored grapes dry

out a little so that their weight decreases by 2%. Let wn be the weight (in kg) of the stored grapes at the

beginning of day n for n 1.

(a) Find a recursive definition for wn. (You may find it helpful to draw a timeline.)

Solution:

A -

AA

weight: w1 w2 w3 wn1 wn

? ? ? ? ?

sold: 2501

2

2502

3

250(n1)

n

? ? ?

day: 1 2 3 n 1 n

decrease: 2% 2% 2%

w1 = 8000 (1 mark)

wn = wn1 0.02wn1 250(n 1)/n

= 0.98wn1 250(n 1)/n for n 2 (3 marks)

(b) Find the value of wn for n = 1, 2, 3.

Solution (2 marks):

w1 = 8000

w2 = 0.98w1 (250 1)/2 = 0.98 8000 125 = 7715

w3 = 0.98w2 (250 2)/3 = 0.98 7715 500/3 = 7394.03

(c) Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of day 1 up

to the beginning of day n for n 1.

Write a MATLAB program to...