CMPSC 465
SOLUTIONS TO ASSIGNMENT 2
This assignment is worth a total of 100 points
Fall 2008
1. [20 Points: solution 18 + simplify 2]
3
n
+3
n

1
8 + 3
n

2
8
2
+3
n

3
8
3
+
. . .
+8
n
is a geometric progression
with ratio
8
3
. Its sum is
next

first
ratio

1
=
8
n
+1
/
3

3
n
8
/
3

1
=
8
n
+1

3
n
+1
5
where the
next
term is the product
last
×
ratio
and the last fraction results from the previous fraction
through multiplication of numerator and denominator by 3.
2. [20 Points: a 3 + b 3 + c 3 + d 3 + e 4 + f 4]
(a)
n
(
n
+ 1)
and
2000
n
2
are both quadratic polynomials and thus have the same growth rate.
(b)
100
n
2
is quadratic and has thus a smaller growth rate than the cubic
0
.
01
n
3
.
(c)
All logarithm functions have the same growth rate.
(d)
log
2
(
n
2
) = 2 log
2
n
. Thus
lim
n
→∞
(log
2
n
)
2
log
2
(
n
2
)
= lim
n
→∞
(log
2
n
)(log
2
n
)
2 log
2
n
= lim
n
→∞
log
2
n
2
=
∞
, which
shows that
(log
2
n
)
2
has a larger rate of growth than
log
2
(
n
2
)
.
(e)
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 Fall '08
 BURCHARDCHARLES
 Logarithm, Mathematics in medieval Islam, brute force algorithm, smaller growth rate

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