MATH INVESTIGATIONS 4 Problem Set: 5 Fall 2013
Teacher (circle): Condie Dover Krouse Pandya
Prince
ID Number____________
Mods: __________
Note: All answers must be supported with written work/explanation. Show sufficient work and
explanation to justify your answers. Remember you are expected to solve problems marked with
without your calculator.
NC
1) Simplify. Show your reasoning.
a) tan   cot   cos 2 
NC
NC
2)
b) csc x  cot x 
sin x
1  cos x
Solve the equation for x in the indicated domain:
2cos2 ( x)  5sin( x)  1 , 0  x  2
n
1 
3) Let an   i  , where i  1 , the imaginary unit. You can use your calculator for this
2 
question.
a) State the first 8 terns of an exactly.
n
b) State the first 6 terms of S n where Sn   an .
i 1
k
1 
c) Compute   i  ,
k 1  2 
10
k
1 

 i  , and
k 1  2 
20
k
1 

 i  accurate to nine decimal places.
k 1  2 
100
d) Without using your calculator, try using the formula for infinite geometric series that we

developed in the sequence and series unit to compute
k
1 

 i  exactly. [Do you think this formula
k 1  2 
might work for series of complex numbers?]
NC
4) Find the area of a triangle with sides of length 13, 14, and 15.
NC
5) The vertices of ABC are the points A(0,6), B(12,0), and C (0,0). A line through the point
(3,0) bisects the area of the triangle. Find the slope of this line.
PS 5.1
Rev. F13
MATH INVESTIGATIONS 4 Problem Set: 5 Fall 2013
Teacher (circle): Condie Dover Krouse Pandya
Prince
ID Number____________
Mods: __________
 1 if n  1,2
6) Consider the sequence: an  
5an  2  2an 1 if n  3
(You may use the sequence mode on your calculator)
a)
b)
c)
State the first 8 terms of the sequence.
a
Let Gn  n 1 . Find G5 .
an
Appproximate lim Gn , accurate to five decmial places.
n 
 5 if n  1

7) Consider the sequence: an   3 if n  2
5a  2a
n 1 if n  3
 n2
a)
b)
c)
NC
State the first 8 terms of the sequence.
a
Let Gn  n 1 . Find G5 . State your answer to five decimal places.
an
Appproximate lim Gn , accurate to five decmial places.
n 
8) Evaluate the following infinite continued fraction: 2 
5
.
5
2
2
5
2
5
The
indicates the fraction continues indefinitely in this pattern. State your answer
exactly and as a five decimal-place approximation. Then ponder whether there is a
relationship with problems 6) and 7)?
9)
 a1  4
a

Consider the sequence:  a2  6
. Let Gn  n 1 . State an infinite continued
an
 a  3a  7 a
n2
n 1
 n
fraction that equals the exact value of lim Gn and use it to calculate this limt.
n 
10)
Write the first 10 terms as reduced fractions of the sequence given by:
1, 1+1, 1 
1
1
1
, 1
, 1
,...
1
1
11
1
1
1
11
1
11
PS 5.2
Rev. F13
MATH INVESTIGATIONS 4 Problem Set: 5 Fall 2013
Teacher (circle): Condie Dover Krouse Pandya
Prince
1a)
1b)
2)
3a)
ID Number____________
Mods: __________
3b)
3c)
3d)
k
1 

 i   .200195313  .400390625i
k 1  2 
10

k
1 

 i   .199999809  .399999619i
k 1  2 
20
k
1 

 i   .200000000  .400000000i
k 1  2 
100
k
i
1 
2
i

[first term  (1  ratio)]

 
1  2i
k 1  2 
i

2i
i (2  i)

(2  i )(2  i)
2i  1

4  i2
1 2
  i
5 5
PS 5.3
Rev. F13
MATH INVESTIGATIONS 4 Problem Set: 5 Fall 2013
Teacher (circle): Condie Dover Krouse Pandya
Prince
4) If we observe that the 13-14-15 triangle can
be divided into two right triangles as shown:
13
ID Number____________
Mods: __________
5)
15
12
5
Then Area =
9
1
1
 5  12   9  12  84
2
2
6a) 1,1,7,19,73,241,847,2899
6b) G5 
7a) 5,3,31,77,309,1003,3551,12117
241
 3.3037
73
6c) Using Calculator, so we guess
7b) G5 
1003
 3.24595
309
b) 7c)
G50  3.4495 and G100  3.4495
lim Gn  3.4495
n 
lim Gn  3.4495
n 
PS 5.4
Rev. F13
MATH INVESTIGATIONS 4 Problem Set: 5 Fall 2013
Teacher (circle): Condie Dover Krouse Pandya
Prince
8) Let,
ID Number____________
Mods: __________
9)
5
x  2
 2
5
2
2
5
x
3
7
5
2
3
x 7
3
7
5
7
 x2  2x  5
 x7
 x2  2 x  5  0
3
3
x
 x2  7 x  3  0
x
22 5
 1  5, since x  0
2
x
7  61
 lim Gn
n 
2
Again, x must be positive!
10)
3 5 8 13 21 34 55 89 144
1, 2, , , , , , , , ,
2 3 5 8 13 21 34 55 89
Holy Fibonacci numbers!
PS 5.5
Rev. F13