# Computational Mathematics

## Computational Mathematics

• 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 = 5tn􀀀1/2, 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
n􀀀1 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/tn􀀀1 =
6 2 (n + 1)! 5n
6 2 (n 􀀀 1 + 1)! 5n􀀀1
=
(n + 1)! 5n
n! 5n􀀀1
=
(n + 1)n! 5  5n􀀀1
n!5n􀀀1
= (n + 1)5
Hence, tn = 5(n + 1) tn􀀀1 for n  2. Therefore, a recursive definition of the sequence (tn) is
t1 = 20; tn = 5(n + 1) tn􀀀1, 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 wn􀀀1 wn
? ? ? ? ?
sold: 2501
2
2502
3
250(n􀀀1)
n
? ? ?
day: 1 2 3 n 􀀀 1 n
decrease: 􀀀2% 􀀀2% 􀀀2%
w1 = 8000 (1 mark)
wn = wn􀀀1 􀀀 0.02wn􀀀1 􀀀 250(n 􀀀 1)/n
= 0.98wn􀀀1 􀀀 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...