Upon amassing a fortune as a rock star, your uncle Zephaniah gives you $1 000 000. You decide to invest the whole lot into Canada Savings bonds (CSB's).
The CSB's pay simple interest at a rate of 3.78% per year, and mature after fifteen years (that is, you have to cash them out after fifteen years).
- Create a chart that shows the total amount of money you will have after each of fifteen years. The headings should be:
"End of Year", "Annual Interest ($) I = Prt" and "Total amount ($)
A = P + I".
- Determine the equation that best models the relationship between the year and the total amount. Explain how you develop this question.
- Use the equation to predict the year that (if you were allowed to leave the bond in for as long as you wanted), your investment would be worth $1 982 800.
- Create a graph that shows the relationship between the year and the total amount the investment is worth.
- Describe the relationship between the mathematics of calculating simple interest, and the matematics behind analysing arithmetic sequences. How might simple interest be described as a problem of applied arithmetic sequences?
In simple interest you pay interest on the original principal only. The amount of interest I is found by `I=Prt` where P is the principal (the initial investment), r the interest rate, and t the number of time periods.
(a) In this problem P=1,000,000 and r=.0378
End of year Annual interest Total amount
1 37800 1,037,800
2 37800 1,075,600
3 37800 1,113,400
etc... (Keep adding 37800 to the final column)
(b) We have a constant rate of change, so we try a linear model. We know the initial amount (at the end of year "0" which is the beginning of year 1 we have 1,000,000) and the slope (the constant rate of change) so the equation is:
so you need approximately 26 years
(d) The graph with time t in years and A in thousands :
(e) In an arithmetic sequence, there is a common difference d. This is analogous to the slope of the line.
The arithmetic sequence has a domain of natural numbers, but the terms of the sequence lie on the line.
The nth term of an arithmetic sequence is `a_n=a_1+(n-1)d` where `a_1` is the first term and `d` is the common difference.
Here we have `a_1=1,037,800` and `d=37800` where `a_1` is the amount at the end of year 1. Thus to find the amount at the end of the year t we have `a_t=1,037,800+(t-1)37,800` or `a_t=1,000,000+37,800t`