In a conversation with a friend, you note that you have two real estate investments, one that has doubled in value in the past 9 years and another that has doubled in value in the past 12 years. Your friend says that the first investment has been growing at approximately 8 % compounded annually and the second at 6 % compounded annually. How did your friend make these estimates? The rule of 72 states that the annual compound rate of growth r of an investment that doubles in n years can be approximated by r = 72 / n . Construct a table comparing the exact rate of growth and the approximate rate provided by the rule of 72 for doubling times of n = 6 , 7 , … , 12 years. Round both rates to one decimal place.
In a conversation with a friend, you note that you have two real estate investments, one that has doubled in value in the past 9 years and another that has doubled in value in the past 12 years. Your friend says that the first investment has been growing at approximately 8 % compounded annually and the second at 6 % compounded annually. How did your friend make these estimates? The rule of 72 states that the annual compound rate of growth r of an investment that doubles in n years can be approximated by r = 72 / n . Construct a table comparing the exact rate of growth and the approximate rate provided by the rule of 72 for doubling times of n = 6 , 7 , … , 12 years. Round both rates to one decimal place.
Solution Summary: The author compares the exact rate of growth and the rate calculated using the rule of 72, for doubling times of n=6,7,mathrm...,12.
In a conversation with a friend, you note that you have two real estate investments, one that has doubled in value in the past
9
years and another that has doubled in value in the past
12
years. Your friend says that the first investment has been growing at approximately
8
%
compounded annually and the second at
6
%
compounded annually. How did your friend make these estimates? The rule of
72
states that the annual compound rate of growth r of an investment that doubles in n years can be approximated by
r
=
72
/
n
. Construct a table comparing the exact rate of growth and the approximate rate provided by the rule of
72
for doubling times of
n
=
6
,
7
,
…
,
12
years. Round both rates to one decimal place.
Keity
x२
1. (i)
Identify which of the following subsets of R2 are open and which
are not.
(a)
A = (2,4) x (1, 2),
(b)
B = (2,4) x {1,2},
(c)
C = (2,4) x R.
Provide a sketch and a brief explanation to each of your answers.
[6 Marks]
(ii)
Give an example of a bounded set in R2 which is not open.
[2 Marks]
(iii)
Give an example of an open set in R2 which is not bounded.
[2 Marks
2.
(i)
Which of the following statements are true? Construct coun-
terexamples for those that are false.
(a)
sequence.
Every bounded sequence (x(n)) nEN C RN has a convergent sub-
(b)
(c)
(d)
Every sequence (x(n)) nEN C RN has a convergent subsequence.
Every convergent sequence (x(n)) nEN C RN is bounded.
Every bounded sequence (x(n)) EN CRN converges.
nЄN
(e)
If a sequence (xn)nEN C RN has a convergent subsequence, then
(xn)nEN is convergent.
[10 Marks]
(ii)
Give an example of a sequence (x(n))nEN CR2 which is located on
the parabola x2 = x², contains infinitely many different points and converges
to the limit x = (2,4).
[5 Marks]
2.
(i) What does it mean to say that a sequence (x(n)) nEN CR2
converges to the limit x E R²?
[1 Mark]
(ii) Prove that if a set ECR2 is closed then every convergent
sequence (x(n))nen in E has its limit in E, that is
(x(n)) CE and x() x
x = E.
[5 Marks]
(iii)
which is located on the parabola x2 = = x
x4, contains a subsequence that
Give an example of an unbounded sequence (r(n)) nEN CR2
(2, 16) and such that x(i)
converges to the limit x = (2, 16) and such that x(i)
#
x() for any i j.
[4 Marks
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