Should I hire a TV or borrow to buy it?

Should I hire a TV or borrow to buy it?

Question

A colour television set is hired for $£80$ in the first year. In each subsequent year the charge is four-fifth of the charge in the previous year. Write down the first four terms of a geometric progression of payments made in successive years, without working them out individually. Calculate the total paid over 8 years by the hirer to the nearest $£$.

If instead he had borrowed $£200$ to buy the set outright and this debt had accumulated at $10\%$ compound interest, what total sum to the nearest $£$ would be owing after 8 years?

Suggestion

How can we sum the terms of a geomtric progression? Is there a formula we can use?

Can we write down how much is owed at the end of year 1? At the end of year 2?

Solution

For simplicity we drop all units as we work and keep in mind that costs are in $£$ and times in years.

The first four payments are $80, 80 \times \dfrac{4}{5}, 80 \times \left(\dfrac{4}{5}\right)^2, 80 \times \left(\dfrac{4}{5}\right)^3.$

These form the first four terms of a geometric progression, with $a = 80,$ and $r = \dfrac{4}{5}.$

The sum of the first $n$ terms of a geometric progression is $S_n = \dfrac{a(1-r^n)}{1-r}$.

We need $S_8 = \dfrac{80(1-0.8^8)}{1-0.8} = £333$ (to the nearest pound).

At the start, he owes $200$. After one year, he owes $200 \times 1.1$ (an increase of $10$ per cent). After $n$ years, he owes $200 \times 1.1^n.$

Thus after eight years, he owes $200 \times 1.1^8 = £429$ (to the nearest pound).

The question doesn’t tell us that the hirer owns the set after eight years, so which option is the best value is moot!