Hi,

In the first year the growth was $42$ centimeters, the growth in the second year was 95% of that or $0.95 \times 42,$ the growth in the third year was thus $0.95 \times 0.95 \times 42 = 0.95^2 \times 42$ and so on. Hence the growth of the tree can be modelled by the geometric sequence

\[a, a r, a r^2, a r^3, \cdot \cdot \cdot , a r^{n-1}, \cdot \cdot \cdot\]

where $a = 42$ centimeters and $r = 0.95.$

Thus the height after 5 years is

\[86 + a + a r + a r^2 + a r^3 + a r^4 \mbox{ centimeters.}\]

To solve the maximum height question I would let $S_n$ be the GROWTH after $n$ years and hence

\[S_n = a + a r + a r^2 + a r^3 + \cdot \cdot \cdot + a r^{n-1}.\]

Multiply both sides of this equation by $r$ to get

\[r S_n = a r + a r^2 + a r^3 + \cdot \cdot \cdot + a r^{n-1} + a r^{n}.\]

Subtracting the second equation from the first yields

\[ (1 - r) S_n = a - a r^{n}\]

or

\[S_n = \frac{a \left( 1 - r^n \right)}{1-r}.\]

You have $r = 0.95$ which is positive and less than 1 so $r^n$ is positive for any positive integer $n,$ and the larger $n$ becomes the closer $r^n$ is to zero. Hence as $n$ gets large, $S_n$ gets closer and closer to $\large \frac{a}{1-r}$ but it never quite gets there.

I hope this helps,
Penny