5/2003

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 x1 to be the data collected the first day, x2 be the data collected the second day, and so on.... and xn 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 {xn }n≥1,

to describe the sequence of numbers where xn is the nth number.

Definition (Range): Consider the sequence {xn }n≥1. The set

{ x1, x2, x3, …….} ={ x2; 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.

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Proposed by Diana Rosu student at St. Thomas Aquinas Secondary School, Brampton, Ontario

Proposed Exercises:

1. 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.
1. 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.
1. 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?
2. A company made a loan for 12 months. The total amount was \$790.60 and they charged \$94.83 interest. Calculate the APR?
3. How many years (to the nearest tenth) will be needed for \$5000 to increase to \$22,000 at 11 percent interest compounded monthly?
4. A 30-year 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 30-year mortgage and how much sooner can it be paid off?
1. 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?
2. 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 tax-free 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?
3. How long will it take for a sum deposited in a savings account to double if it is paid 6 percent interest compounded quarterly?
4. 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?