Talk is cheap... because supply exceeds demand.
 Anonymous
Theory
Sequences
Sequences
are to calculus what at calculator is to a scientist. There are many
ways to introduce sequences. Here we will follow a somewhat unorthodox
way. Indeed, consider a scientist doing an experiment; he is
collecting data, let us say, every day.
So,
put x_{1} to be the data collected the first day, x_{2 }be
the data collected the second day, and so on.... and x_{n }is
the data collected after n days. Clearly, we are generating a set of
numbers with a very special characteristic: there is an order on the
number, that is, we naturally have the first number, the second
number, and so on....
A
sequence is by definition a set of real numbers with this natural
order. We wil use the notation {x_{n }}_{n≥1},
to
describe the sequence of numbers where x_{n} is the nth
number.
Definition
(Range):
Consider the sequence {x_{n }}_{n≥1}. The set
{ x_{1,
}x_{2, }x_{3, …….} ={ }x_{2}; n =
1,2,3,…}
is
called the range of the sequence.
Of
course, in the range there is no order. For example, consider the
sequence {(1)}_{n≥1}. Its range is the set {1, 1}. It
has two elements. The sequence itself is alternating between 1 and 1.
Solutions
from the Previous Issue:
1
Solution:
2
Solution:
3
Solution:
4
Solution:
5
Solution:
6
Solution:
7
Solution:
8
Solution:
9
Solution:
10
Solution:
Proposed by Diana Rosu student at St.
Thomas Aquinas Secondary School, Brampton, Ontario
Proposed Exercises:
 I
want to calculate the amount of money I will have in 29 years if I
put $120 per month into an account that will earn 15 percent
interest per year.
 Brian
makes $2500 deposits into his Registered Retirement Savings Plan
on his 20th, 25th, 30th, 35th, and 40th birthdays. The RRSP pays
7% per annum, compounded annually. Determine the value of Brian's
RRSP when he's 65.
 If
Li Fong had earned one fourth of a percent more in annual interest
on an investment, the interest for one year would have been $45
greater. How much did she invest at the beginning of the year?
 A
company made a loan for 12 months. The total amount was $790.60
and they charged $94.83 interest. Calculate the APR?
 How
many years (to the nearest tenth) will be needed for $5000 to
increase to $22,000 at 11 percent interest compounded monthly?
 A
30year home mortgage is taken out for $161,800 at an interest
rate of 7.5%. The first payment on the loan is made in December,
in the amount of $1,232.65. If one extra payment is made each
year, for the same amount as a regular payment, how much money can
be saved over the life of the 30year mortgage and how much sooner
can it be paid off?
 At
what rate of simple interest would the amount to be repaid on a
loan be triple the principal of the loan after 25 years?
 Nancy
Dunn has inherited $80,000 from her uncle. She invests part of
the money in a video rental firm which produces a return of
7 percent per year, and divides
the rest equally between a taxfree bond at 6 percent a year and
money market fund at 12 percent a year.
Her annnual return on these investments is $6800. How much is invested in each?
 How
long will it take for a sum deposited in a savings account to
double if it is paid 6 percent interest compounded quarterly?
 If
Martin started with a company at an annual salary of $18,000, and
he gets an increase of 5 percent at the end of every year, at this
growth rate, what would Martin's salary be in his 30th year with
his company?
