Lesson

We are quite familiar with compound interest and the associated calculations from work we've studied previously.

Here we'll take a look at various way to model and represent compound interest so that we may clearly examine what is happening over time.

We can model compound interest in two main ways:

- Using a recurrence relation
- Using a table of value or spreadsheet

Using a recurrence relation allows us to progressively analyse the effect of compounding our investment at each time period. In simple situations like those we'll examine here, we could also use the general rule for a sequence.

Let's take the simple example of investing $\$1000$$1000 at $10%$10% interest compounded annually.

What will the first few years of our investment look like?

End of Year 1 | $1000\times1.10=\$1100$1000×1.10=$1100 |
---|---|

End of Year 2 | $1100\times1.10=\$1210$1100×1.10=$1210 |

End of Year 3 | $1210\times1.10=\$1331$1210×1.10=$1331 |

We can clearly see that the balance at the end of each year is 1$10%$10% of the previous year.

Expressing this as a recurrence relation where $B_{n+1}$`B``n`+1 is the balance at the end of year $n$`n`, we have:

$B_{n+1}=1.1B_n;B_0=1000$`B``n`+1=1.1`B``n`;`B`0=1000

Notice the use of $B_0$`B`0 here to indicate our initial investment. If instead we wanted to use $B_1$`B`1, we need to be mindful that $B_1$`B`1represents the balance at the end of year 1 and so we could write our recurrence relation as:

$B_{n+1}=1.1B_n;B_1=1100$`B``n`+1=1.1`B``n`;`B`1=1100

I prefer to use $B_0$`B`0 since writing it this way doesn't require me to do any calculations!

Recall that this recurrence relation represents a geometric progression, so we could also write a general or explicit rule for this scenario.

We could have $T_n=1100\times1.1^{n-1}$`T``n`=1100×1.1`n`−1 and notice we have to use $T_1=1100$`T`1=1100 in this case.

Or we could use $T_n=1000\times1.1^n$`T``n`=1000×1.1`n`

I encourage you to be flexible with how you write and interpret your recursive and general rules - it will make your progress with this work so much easier!

Let's explore this interactive compound interest spreadsheet.

You can change the amount invested (the blue cell) to any value you'd like to invest.

You can change the annual interest rate (the green cell) to any value.

You can change the number of compounding periods (the pink cell) to quarterly ($4$4), monthly ($12$12), weekly ($52$52) or perhaps daily ($365$365).

What happens as you increase the number of compounding periods?

What happens as you increase the annual interest rate?

How has the value in cell C10 been calculated?

How has the value in D12 been calculated?

When you can answer these questions you're ready for the worked examples below.

$£2000$£2000 is invested at the beginning of the year in an account that earns $4%$4% per annum interest, compounded quarterly.

How much money is in the account at the end of the first year?

Give your answer to the nearest penny.

Write a recursive rule, $V_n$

`V``n`, that gives the balance in the account at the end of the $n$`n`th quarter.Write both parts of the rule (including for $V_0$

`V`0) on the same line, separated by a comma. Express all necessary values as decimals.

The following spreadsheet shows the balance (in pounds) in a savings account in 2014, where interest is compounded monthly.

A | B | C | D | |

1 | Month | Balance at beginning of month | Interest | Balance at end of month |

2 | July | $8000$8000 | $160$160 | $X$X |

3 | August | $8160$8160 | $163.20$163.20 | $8323.20$8323.20 |

4 | September | $8323.20$8323.20 | $Y$Y |
$8489.66$8489.66 |

5 | October | $Z$Z |
$169.79$169.79 | $8659.45$8659.45 |

6 | November | $8659.45$8659.45 | $173.19$173.19 | $8832.64$8832.64 |

Calculate the value of $X$

`X`.Use the numbers for July to calculate the monthly interest rate.

Calculate the value of $Y$

`Y`.Calculate the value of $Z$

`Z`.Write a recursive rule, $B_n$

`B``n`, that gives the balance at the end of the $n$`n`th month, with July being the first month.Write both parts of the rule (including for $B_0$

`B`0) on the same line, separated by a comma.Write an explicit rule for $B_n$

`B``n`, the balance at the end of the $n$`n`th month, with July being the first month.