Mathematical Investigations IV S&S Unit test Name ______________
You may use a TI-30 Calculator on this exam. Justify all your work.
1.
(2 pts each) Given the numbers 12 and 48, find their:
b) geometric mean c) harmonic mean a) arithmetic mean
2. (4 pts) Consider the series
4

12

36

108

7 13 19 25

8748
49
.Write the series in
  notation.
3.
(5 pts) Use appropriate formulas to evaluate the series k
79 

5
( k

1)
2 .

Mathematical Investigations IV S&S Unit test Name ______________
4. (4 pts) The fifth term in an arithmetic sequence is 9 and the 23 rd term is 54. Find the sum of all 23 terms.
5.
(2 pts each) If a n

3
 

2
3 a n

1 if n

1 if n

1
, a) State the first five terms of the sequence using common fractions. b) Give an explicit formula for the sequence.
6. (3 pts) Write out the first four terms of the series k
92 

1
2 k 
4
3 k

Mathematical Investigations IV S&S Unit test Name ______________
7. (4 pts) Let
S n
 explicit formula for k n 
.
Suppose that 2
2

1 a k
S n
 n
  a k
, k

2
.
for all positive integers n . Determine an
8. (4 pts) A certain infinite geometric series has only positive terms and its sum is equal to
8
times the arithmetic mean of the first two terms. What is the common ratio
3 of the series.
9. (3 pts) Evaluate the following infinite continued fraction:
2

2

5
5
.
2

2

5
5

Mathematical Investigations IV S&S Unit test Name ______________
9. Let S represent the n n th
partial sum of the series k



1 ln a.
(4 pts) Use mathematical induction to prove that S n k k

1
.
  ln( n

1) for all positive integers n . b.(2 pts) Determine whether the series k



1 ln k k

1
converges or diverges. If it converges, find its value. You may use the formula for the partial sum given in part a..