Math 1090-004
Midterm 3
April 24, 2015
Please show all of your work as partial credit will be given where
appropriate, and there may be no credit given for problems where there is no
work shown. You may use a scientific calculator and a 4 × 6 inch note card.
Scratch paper will be provided but NOT collected, so please transfer all
finished work onto the proper page in the test. This exam totals 70 points.
Name:
UID:
Formulas
i If a1 , a2 · · · is an arithmetic sequence where at = at−1 + d, then a1 + a2 + · · · + an =
n
(a1 + an ).
2
n
−1
ii If a1 , a2 · · · is a geometric sequence where at = dat−1 , then a1 + a2 + · · · + an = a1 dd−1
.
iii Simple interest: S = P (1 + rt).
nt
iv Compound interest: R = P 1 + nr .
n
v AP Y = 1 + nr − 1 or AP Y = er − 1.
R((1+rc )N −1)
, where rc = nr , N = nt.
vi Future value of an ordinary annuity S =
rc
i
h
N +1
vii Future value of an annuity due: S = R (1+rc )rc −1 − R, where rc = nr , N = nt.
R(1−(1+rc )−N )
,
rc
where rc = nr , N = nt.
R(1+rc )(1−(1+rc )−N )
,
rc
where rc = nr , N = nt.
viii Present value of an ordinary annuity: S =
ix Present value of an annuity due: S =
x Present value of a deferred ordinary annuity: P =
−(N −k)
xi Amortization formula: SN −k = R 1−(1+rrcc)
.
R(1−(1+rc )−N )
.
rc (1+rc )m
1. (10 pts) Solve the equation
2x+1
x−5
=4−
3
x−5
2. (10 pts) Given a geometric sequence with a1 =
10 terms of the sequence.
3
2
and a6 =
3
,
64
find the sum of the first
It suffices to write down the final expression that you would plug into your calculator.
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3. (10 pts) Solve and graph the solutions of the following system of inequalities
x + 7y ≤ −15
5x − y ≥ −3
x − 2y ≤ 12
You need to: graph the lines, find the coordinates of the vertices of the resulting region, and
shade it.
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4.
(a) (5 pts) How much should be invested quarterly (at the end of each quarter) at 12%
compounded quarterly to pay off a debt of $30, 000 in 6 years?
Start by writing down the formula that you will be using.
(b) (5 pts) If we invest $700 into the account in (a) at the end of each quarter, how many
years will it take to pay off the debt?
Start by writing down the formula that you will be using. It suffices to write down the
final expression that you would plug into your calculator.
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5.
(a) (3 pts) What is the annual percentage yield (APY)? Write a sentence explaining this,
not a formula.
(b) (7 pts) Which is the better investment deal?
(i)
(ii)
(iii)
(iv)
An
An
An
An
account
account
account
account
earning
earning
earning
earning
10% compounded annually.
9% interest compounded quarterly.
8.6% interest compounded monthly.
8% interest compounded continuously.
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6. (10 pts) George and Mary have a friendly contest between them to see who can earn the most
money in the next 5 years. They agree not to invest more than $10, 500 each in an account
earning 10% interest compounded monthly, but neither of them has that much money up front.
George decides to invest $5, 000 at the beginning, and then add $85 at the end of each month
to his account. Mary invests $170 at the end of every month for the entire 5 years. Answer
the questions below to see who has more money at the end of the 5 years.
(a) (1 pt) How much money did George and Mary invest?
(b) (5 pts) How much money does George have at the end of the 5 years?
Start by writing down the formula(s) that you will be using.
(c) (4 pts) How much money does Mary have at the end of the 5 years?
Start by writing down the formula(s) that you will be using.
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7. For the following 3 questions, you don’t need to use your calculator: it suffices to write down
the final expression that you would type in. Make sure to always indicate which formula(s)
you are using.
(a) (3 pts) How much will an inheritance of $100, 000 provide at the end of each month for
10 years if money is worth 7.2% compounded monthly?
(b) (4 pts) How much will an inheritance of $100, 000 provide at the end of each month for
10 years if money is worth 7.2% compounded monthly and we don’t start withdrawing
until the 5th month of the 3rd year?
(c) (3 pts) How much will an inheritance of $100, 000 provide at the beginning of each month
for 10 years if money is worth 7.2% compounded monthly and we don’t start withdrawing
until the 5th month of the 3rd year?
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