IMSA
MI-4
Prob. Set #4
DUE: Friday, September 18
Fall 2015
1)
Robin has a hobby of building dollhouse furniture. A model piano consumes 1.5 ml of paint and
uses 25 cc of balsa wood. His good friend Madeline would like Robin to make one for her
dollhouse which is scaled twice as large as Robin's. How much paint and how much wood will it
take for him to produce a model piano that has twice the dimensions of his model?
2)
Simplify: 3+
2
2
3+
3+
2
3+
2
where … indicates that the pattern continues indefinitely. Give exact answer.
n
The notation   , read "n choose r", is the number of combinations of n objects
r
taken r at a time. That is, it is the number of ways to select a group of r objects
from a set of n distinct objects. It is equal to
n!
n(n -1) (n - r +1)
=
r !(n  r )!
r(r -1) 2×1
where n and r are non-negative integers and n  r. We often write C(n,r) or nCr
to denote combinations as well. It is also called a binomial coefficient.
3)
16   16 
NC a) Solve for n, if    
.
 n   n  2
 n  2   n  3  n  4 
NC b) 


  109 .
 2   2   2 
4)
Find all values of n, 0  n  30 such that
 k 
a)  sin 
0
 6 
k 0
n
n
b)
 k 
0
4 
 cos 
k 0
PS 4.1
Rev. F15
IMSA
5)
MI-4
Prob. Set #4
Fall 2015
In the following pattern, how many dots are there in the nth figure?
n=1
6)
DUE: Friday, September 18
Consider a sequence
Show that
n=2
n=3
¥
{ an }n=1 such that
a1  a2 
¥
{ an }n=1 is an arithmetic sequence.
n=4
 ak  k 2  3k for k  1, 2,3
.
Find a recursive formula for an .
2n
7)
Find a simplified formula, in factored form, for:
 k  k  1 using the formulas you learned in the
k 1
sequences and series unit.

8)
Evaluate the following sums:
a.

k 1
 2k  1
2k

b.
k2
.

k
k 1 2
Though we do not have formulae for these types of sums from our classroom work, the methods
that we learned should help to evaluate the sums. You may assume that both sums converge.
9)
If cos   x and 0   
a) sin 

2
, find exact values, in terms of x, for each of the following:
b) tan(- b )
c) sin ( p – b )
d) sec ( b – p )
10) Find the domain for each of the following:
1
a) f ( x)  2 x
b) h( x)  log 4  x 2  9 x  19 
e  5e x  4
PS 4.2
Rev. F15
IMSA
MI-4
Prob. Set #4
DUE: Friday, September 18
Fall 2015
11) Consider the graph below, which is the graph of y = a tan(bx) for some choice of a and b.


a) If the graph contains the point  ,  3 3  determine a and b.
3

b) What is the domain of the function?
a1  1

12) Consider the sequence: a2  1
a  2a  3a
n2
n 1
 n
a) State the frist 8 terms of the sequence.
a
b) Let Gn  n 1 . Find G5 .
an
c) Give a five decimal appproximation of lim Gn . Hmm, this answer looks familiar.
n 
13) Find real numbers A, B, and C such that
26 x 2  53x  44
A
B
C


 .
2
2
3x( x  2)
x  2 ( x  2) 3x
PS 4.3
Rev. F15