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...

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).

 

  1. 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".
  2. Determine the equation that best models the relationship between the year and the total amount. Explain how you develop this question.
  3. 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.
  4. Create a graph that shows the relationship between the year and the total amount the investment is worth.
  5. 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?

 

 

Asked on by moeenmoeen

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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:

`A=1,000,000+.0378t`

(c) 1,982,800=1,000,000+.0378t

982,800=.0378t

`t~~25.978835`

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`

Sources:

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