Geometric progression question. Year less?

Question

Here is the math 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?

This is how I worked out the question.

100*(5/7)^10

Which is 3.46m to 2d.p.

However my teacher said that it is wrong (to the whole class and also insulted me a bit >_>) and that I didn't follow this formula.

Tn = ar^(n-1)

And that I should have done

100(5/7)^9

Which is instead 4.84m to 2d.p.

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This doesn't make sense because if you used the formula and the question was

What will the height of the cliff be after 1 year?

 100(5/7)^(1-1)
=100(5/7)^0
=100

That doesn't make sense at all!

Am I right or is the teacher right?

ADDITIONALLY my teacher said his answer is an interpretation of the question. Is his answer a valid interpretation of this question? Or is it just incorrect mathematics?

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BTW Also my teacher said my answer has no common sense and that I won't be able to do the HSC well if I keep reading questions wrong.

Answer 1

Well, we agree that $T_{n+1} = (5/7) T_n,$ so $T_n = T_0 (5/7)^n$. Clearly, after $0$ years, the height of the tree is $100$, so you have $T_n = 100 (5/7)^n$.

Hence, after one year, we have $$ T_1 = 100 (5/7) $$ and after 10 years, we have $$ T_1 = 100 (5/7)^{10} $$ as you have claimed.

Answer 2

$T_n = ar^n-1$

$T_1 = 100\left(\frac57\right)^1-1= 100m $

So $T_{10}=100\left(\frac57\right)^{10}-1= 4.84m $

So your teacher is correct