MATH INVESTIGATIONS 4 Problem Set: 5 Fall 2015
Teacher (circle): Condie
Prince
Stalmack
ID Number____________
Mods: __________
As usual, unless otherwise indicated, calculator use should be limited to evaluating functions (i.e., no
“solve” command; figure out answers “by hand” and use calculator to get numeric answers).
1)
Solve each equation for n:
 2n 
 2n  1
a) 13  
  7

 n  2
 n3 
 3n  1  3n  1
b) 4  


 n   n 1 
n
2)
1 
Let an   i  , where i  1 , the imaginary unit. You can use your calculator for this question.
2 
a) State the first 8 terns of an exactly.
n
b) State the first 6 terms of S n where S n   an .
i 1
k
1 

 i ,
k 1  2 
20
c) Compute
k
1 

 i  , and
k 1  2 
200
k
1 

 i  accurate to nine decimal places.
k 1  2 
1000
k
1 
d) Compute   i  exactly (show non-calculator work!).
k 1  2 

.
3)
4)
Find, to the nearest hundredth of a degree, the acute angle formed by two of the “main” diagonals
of a cube.
a1  1

(Feel free to use your calculator’s sequence mode.) Consider the sequence: a2  1
a  5a  2a
n2
n 1
 n
a) State the first 8 terms of the sequence.
a
b) Let Gn  n 1 . Find G5 .
an
c) Appproximate lim Gn , accurate to five decmial places.
n 
PS 5.1
Rev. F15
MATH INVESTIGATIONS 4 Problem Set: 5 Fall 2015
Teacher (circle): Condie
Prince
Stalmack
5)
ID Number____________
Mods: __________
a1  5

Consider the sequence: a2  3
a  5a  2a
n2
n 1
 n
a) State the first 8 terms of the sequence.
a
b) Let Gn  n 1 . Find G5 . State your answer to five decimal places.
an
c) Appproximate lim Gn , accurate to five decmial places.
n 
6)
Evaluate the following infinite continued fraction: 2 
5
.
5
2
2
5
2
5
State your answer exactly and as a five decimal-place approximation. Then ponder
whether there is a relationship between problems 5) and 6)?
7)
a1  4
a

Consider the sequence: a2  6
. Let Gn  n 1 . State an infinite continued fraction
an
a  3a  7a
n2
n 1
 n
that equals the exact value of lim Gn and use it to calculate this limt.
n 
8)
Write the first 10 terms as reduced fractions of the sequence given by:
1, 1+1, 1 
9)
1
1
1
, 1
, 1
,...
1
1
11
1
1
1
11
1
11
Suppose that an is a sequence of positive numbers such that an1  an for all n (we say an is
strictly increasing). If 250 and 2015 are terms in this sequence and the sequence satisfies an  an1  an2
for all n  3 . Find the minimum value of a1 .
PS 5.2
Rev. F15