In Simple Terms
Annie and Ben are both doing the question...
A 100m cliff erodes by 2/7 of its height each year.
(a) What will the height of the cliff be after 10 years?
Annie believes the answer is 100*(5/7)^9 because using the geometric progression formula exactly is common sense.
Ben believes the answer is 100*(5/7)^10 because it's 10 years later.
In your answer, determine whether Annie is right or wrong, giving reasons. Also determine whether Ben is right or wrong, also giving reasons.
The Actual Question
An extension to this question.
Can the question...
A 100m cliff erodes by 2/7 of its height each year.
(a) What will the height of the cliff be after 10 years?
...Be interpreted in different ways so that you could answer like this:
100*(5/7)^10
=976562500/282475249
=3.46m to 2d.p.
Or this:
100*(5/7)^9
=195312500/40353607
=4.84m to 2d.p.
I tried to convince my teacher that his answer wasn't correct but he said that his answer was an interpretation. The answer made in this previous question supported my answer, but my teacher still thinks he is right.
Can you really interpret this question in the 2 ways above or is there only just one right and the rest wrong? (Excluding the simplification of answers. Also some other questions are designed to have more than one answer.)
This is the only interpretation:
After one year, the height is $100\times (5/7)^1$.
After two years, the height is $100\times (5/7)^2$.
...
After ten years, the height is $100\times (5/7)^{10}$.
Now if we use the geometric formula you mentioned, after ten years, the term we look at is actually the eleventh term, and so the formula also gives $100\times (5/7)^{11-1}$.