Problem Set 6 Solutions - Problem Set 6 Solutions Due:...

Problem Set 6 Solutions
Due: Monday, March 28 at 9 PM
Problem 1. Sammy the Shark is a financial service provider who offers loans on the fol-
lowing terms.
• Sammy loans a client m dollars in the morning. This puts the client m dollars in
debt to Sammy.
• Each evening, Sammy first charges a “service fee”, which increases the client’s debt
by f dollars, and then Sammy charges interest, which multiplies the debt by a factor
of p . For example, if Sammy’s interest rate were a modest 5% per day, then p would
be 1 . 05 .
(a) What is the client’s debt at the end of the first day?
(b) What is the client’s debt at the end of the second day?
Problem 2. Find closed-form expressions equal to the following sums. Show your work.
(c) Write a formula for the client’s debt after d days and find an equivalent closed
form.

�
�
�
2 Problem Set 6
(a)
n �
9 i − 7 i
11
i
i =0
Solution. Split the expression into two geometric series and then apply the formula
for the sum of a geometric series.
n n
� �
i
n
�
�
9 i − 7 i
�
9
�
7
�
i
=
11
i
11
−
11
i =0 i =0 i =0
�
9
�
n +1
�
7
�
n +1
1 −
11
1 −
11
=
9 1 −
11
−
� �
n +1
7 1 −
11
� �
n +1
11 9 11 7 11
= + + −
2
·
11 4
·
11 4
(b)
n
3
4 i +5
i =1
Solution. Taking the logarithm reduces this product to an easy sum.
n
3
log
3
(
Q
n
3
4 i +5
)
3
4 i +5
=
i =1
i =1
P
n
= 3
4 i +5
i =1
= 3
2 n ( n +1)+5 n
(c)
n ∞
�
� �
1
�
i
j
5 / 3
· 1 −
2 j
1 / 3
j =1 i =0
Solution. This fearsome-looking sum is a paper tiger; we just apply the formula for
the sum of a geometric series followed by the formula for the sum of an arithmetic
series.
n
� �
�
1
�
i
�
n ∞
1
j
5 / 3
· 1 −
2 j 1 / 3
= j
5 / 3
·
�
1
�
j =1 i =0 j =1
1 − 1 −
2 j 1 / 3
n
= 2 j
2
j =1
1
2 n ( n +
2
)( n + 1)
=
3

3 Problem Set 6