# Find the maximum height that the Pohutukawa tree is expected to reach in cm?

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A Pohutukawa tree is 86 cm tall when it is planted. In the first year after it is planted, the tree grows 42cm in height. Each year the tree grows in height by 95% of the growth of the previous year.

A Pohutukawa tree is 86 cm tall when it is planted. In the first year after it is planted, the tree grows 42cm in height. Each year the tree grows in height by 95% of the growth of the previous year.

##### 3 Answers

Height after 5 years: 276cm

**Edit**

Maximum height: 926cm.

#### Explanation:

The growth of the tree over n years is

The sum of a geometric progression is,

Therefore the height in 5 years is 190.02cm + the initial 86cm = 276cm.

**Edit** I see that you have changed the question to ask about the maximum height of the tree. In this case, the formula

Added to the initial height of 86 cm, gives 926cm.

926cm

#### Explanation:

This is going to need a double check...

The tree starts at 86cm. Year one, the tree will be:

Year two, the tree will be

Year three the tree will be

This goes on year after year. One of the things we can do is factor out the 42, so our tree looks like this:

All of those (.95) terms (even the 1) can be written as exponents of (.95) so:

If you calculate the summation of the (.95) exponential terms, you get 20

Therefore, the maximum height of the tree (H) will be:

#### Explanation:

The general formula for a converging geometric series is

where

and

In this case

So the final (maximum) height will be