Download Jeopardy Review for EXAM 4 - University of Arizona Math

MATH 110
EXAM 4 Review
Arithmetic
sequence
Geometric
Sequence
Sum of an
Arithmetic
Series
Sum of a Finite
Geometric
Series
Sum of Infinite
Geometric Series
Jeopardy
Number
Leftovers!? Patterns
“Sum”thing
Two Steps Giant
Back
Leaps
Forward
Potent
Potables
100
100
100
100
100
100
200
200
200
200
200
200
300
300
300
300
300
300
400
400
400
400
400
400
500
500
500
500
500
500
Leftovers !? 100
• A culture of bacteria originally numbers
500 spores. After 2 hours there are 1500
bacteria. Assuming the number of spores
can be modeled by the exponential
function N  N 0e kt determine how many
spores will be present in 6 hours.
• Answer: 13,500 spores
Leftovers !? 200
• The cost of tuition at four-year public universities
has been increasing roughly exponentially for
the past several years. In 1997, average tuition
was $3,111 while in 2004 it was $5,132.
Assuming that tuition will increase according to
t
P

P
a
the exponential model
, at what rate is
0
tuition increasing each year.
• Answer: 7.41%
Leftovers !? 300
• When rabbits were first brought to Australia, they
multiplied very rapidly because there were no
predators. In 1865, there were 60,000 rabbits.
By 1867, there 2,400,000 rabbits. Assuming
exponential growth according to the model
kt
P  P0e , when was the first pair of rabbits
introduced into the country?
• Answer: Around 1859
Leftovers !? 400
• The consumer price index compares the cost of
goods and services over various years. The
base year for comparison is 1967. The same
goods and services that cost $100 in 1967 cost
$184.50 in 1977. Assuming that costs increase
exponentially according to P  P0e kt , when did
the same goods and services cost double that of
1967?
• Answer: 1978
Leftovers !? 500
• The proportion of carbon-14, an isotope of
carbon, in living plant matter is constant. Once a
plant dies, the carbon-14 in it begins to decay
with a half-life of 5570 years. An archaeologist
measures the remains of carbon-14 in a
prehistoric hut and finds it to be one-tenth the
amount of carbon-14 in the living wood. How
old is the hut?
• Answer: 18,503 years old
Number Patterns 100
• Consider the following sequence:
7, 11, 15, 19, . . .
Find the 150 term.
• Answer: 603
Number Patterns 200
• How many terms are there in the following
sequence:
7 11 15
103
,  , , ,
2 5 10
626
• Answer: 25 terms
Number Patterns 300
• List the first four terms for the sequence
whose formula is:
 5
 1
an  
 an 1
n 1
n  2,3,4,
• Answer: 5, (1/5), 5, (1/5)
Number Patterns 400
• Consider the following sequence:
375, -75, 15, -3, . . .
• What is the sign of the 493 term?
• Determine the most evident formula for the nth
term of the sequence.
• Answer: positive;
 1
an  375  
 5
n 1
Number Patterns 500
• Write a formula for the following sequence:
• 1, 3, 6, 10, 15, . . .


n
n

1
• Answer: an 
2
“Sum”thing 100
4
• Evaluate the expression:
 20  m!
m 0
• Answer: 66
“Sum”thing 200
• Re-index the following summation so that
it starts at k = 1:
 k
100
k 17
 5   k  32k  261
84
2
2
k 1
“Sum”thing 300
• It can be shown that the Euler number, e, can be
approximated taking the square root of the
following series:
2 4 8
512
1   
1! 2! 3!
362880
• Write this series using sigma notation.
2k 1
• Answer: 
k 1 k  1!
10
“Sum”thing 400
• Find the
first term given that
25
and  ak  75
k 1
• Answer: 25
25
a
k 2
k
 50
“Sum”thing 500
• Find the value of the sum:
 k
100
k 17
• Answer:
337,274
2
 5
Two Steps Back 100
• Find the common difference of an
arithmetic sequence whose 16th term is 73 and whose 21st term is -103.
• Answer: -6
Two Steps Back 200
• Find the twentieth term of the arithmetic
sequence whose third term is 6 and whose
sixth term is 18.
• Answer: 74
Two Steps Back 300
• Which of the following sequences is
arithmetic?
(I) 2,2 5,10,10 5,50,
(II) 4,4  3 ,4  6 ,4  9 ,4  12 ,
3 17
(III)  2, ,5, ,12,
2
2
Two Steps Back 400
• Find the sum of the first 30 terms in an
arithmetic sequence where the 6th term is
7 and the 12th term is 12.
• Answer: 447.5
Two Steps Back 500
• Find the sum of all the numbers in the
sequence:
4,13,22,31,580
• Answer: 18980
Giant Leaps Forward 100
• Find the common ratio of a geometric
sequence where the 4th term is 100 and
the 7th term is 4/5.
• Answer: 1/5
Giant Leaps Forward 200
• Find the first term of a geometric
sequence whose fourth term is -8 and
whose tenth term is -512/729.
• Answer: -27
Giant Leaps Forward 300
• Which of the following sequences is
geometric?
(I) 3 2 ,6,6 2 ,12,12 2, 
(II) 192,48,12,3,
(III) zb , zb  2, zb  4, zb  6, zb  8,
Giant Leaps Forward 400
• Find the indicated sum:
7
28  14  7   
2
• Answer: 56/3
Giant Leaps Forward 500
• A company plans to contribute to each of its employees’
retirement fund by depositing $100 at the end of each
month in a retirement account. The account pays 6%
interest compounded monthly. A look at the account
balance shows that the amount is a series:
2
3
 0.06 
 0.06 
 0.06 
 0.06 
1001 

100
1


100
1




100





1 

12
12
12
12








• How much money will there be after 18 years
• Answer: 38,929
216
Potent Potables 100
• Find the sum:
 2
10
k 1
• Answer:
1661
k
k
2

Potent Potables 200
• Find the sum of the infinite geometric
series if possible:
9  12  16  
• Answer: Not possible
Potent Potables 300
• A repeating decimal can always be
expressed as a fraction. Consider the
decimal: 0.23232323… Use the fact that:
0.23232323… = 0.23 + 0.0023 + 0.000023
+ ….
To write 0.232323… as a fraction.
• Answer: 23/99
Potent Potables 400
• This problem illustrates how banks create credit and can
lend out more money than has been deposited.
Suppose that $100 is deposited in a bank. Experience
shows that on average on 8% of the money deposited is
withdrawn by the owner, which means that bank are free
to lend 92% of their deposits. Thus, $92 of the original
$100 is loaned out to other customers. This $92 will
become someone else’s income, and eventually will be
redeposited in the bank. So $92(0.92) =$84.64 is loaned
out again and then redeposited, and so on. Find the
total amount of money deposited in the bank.
• Answer: $1250
Potent Potables 500
2
• Solve for x: 1  x  x  x  x   
3
2
• Answer: 1/2
3
4