Mathematical Investigations III
Name ___
Mods:
Sequences and Series
You may use a TI-30 Calculator on this exam. Justify all your work.
n
n
n(n  1)(2n  1)
n 2 (n  1) 2
2
3
Useful formulas:
j

j



6
4
j 1
j 1
True or False:
_____ 1. 1  3  5  7...  35  182
_____ 2. The sequence 2, 2, 2, 2, 2, . . . is arithmetic, geometric, and harmonic.
_____ 3. 219 is a term of the geometric sequence beginning 1, 2, 4, 8, 
_____ 4. The harmonic mean of 1, 1 , and 2 is
1
.
6
_____ 5. If Bert can mow the lawn in 2 hours and Ernie can mow it in three hours, then
it will take them 2.5 hours if they work together.
_____ 6.
12
8
i 5
i 1
  2i  1    2i  9 
1 1 1 1
_____ 7. The sequence 1,  , ,  , ,
2 3 4 5
converges to 0.
_____ 8. 0.2, 0.22, 0.222, 0.2222, 0.22222, . . . is a geometric sequence.
1 1 1 1
2
_____ 9. 1      ... 
2 4 8 16
3
10. Find recursive and explicit definition for the sequence beginning
8
4 2
,  , , 1,
27 9 3
_
11. Expand, but do not calculate,
7
 (n
2
 n) .
n 3
 1
12a) Find  3   
2
k 1 
5
b) Find

k 1

k 1
 1
3  

2
k 1 
 1
c) Find  3   
2
k 6 
k 1
using the formula for a finite geometric series.
.
.
d) What relationship exists between your answers to (a)-(c), and why?
Sequences and Series Test
8 February 2010
13. Find the sum
100
 (4k  2) .
k 1
14. Evaluate 5  4.9  4.8 
 1.2 1.1 1.
15. Find   2i 2  3 . State your answer in terms of n.
n
i 1
a1  3
16.
an 
2
an 1
3
a) State the first five terms of the sequence using common fractions.
b) Give an explicit formula for the sequence.
Sequences and Series Test
8 February 2010
17. Evaluate


j 2

1
1

 2 j 1  2 j  5  .

18. Suppose you have two one-liter flasks, A and B. Initially, flask A is completely
filled with water, while B is empty. In the first minute, you pour (without spilling!)
one-third of the contents of flask A into flask B. Each minute thereafter, you pour onethird of the remaining contents of flask A into flask B.
Now let an and bn represent the amount of water, in liters, in flasks A and B after n
minutes.
a) Write a recursive definition for the sequence an .
b) Suppose the sequence cn is defined by cn  an  bn . Find a formula for cn .
c) Suppose S n is the sequence of partial sums for bn . Find S 3 .
d) After how many minutes will flask A be empty?
Sequences and Series Test
8 February 2010
Sequences and Series Test
8 February 2010