Question

The height of a tree 3 meter and increased 2 meter in the course of the next year, if in each succeeding year, the growth is 2÷3 of the year before, find limiting height of th tree

Answer 1

The growth pattern of the tree can be represented as a geometric sequence, where each term is 2/3 of the previous term. The initial height is 3 meters, and it increases by 2 meters in the first year.

Let's denote the height after each year as follows:

- Year 1: \(3 + 2 = 5\) meters

- Year 2: \(5 \times \frac{2}{3} = \frac{10}{3}\) meters

- Year 3: \(\frac{10}{3} \times \frac{2}{3} = \frac{20}{9}\) meters

- ...

The limiting height (\(H_{\text{limit}}\)) of the tree can be found as the sum of an infinite geometric series with the first term (\(a\)) of 5 meters and a common ratio (\(r\)) of 2/3:

\[ H_{\text{limit}} = \frac{a}{1 - r} \]

\[ H_{\text{limit}} = \frac{5}{1 - \frac{2}{3}} \]

Now, calculate \(H_{\text{limit}}\):

\[ H_{\text{limit}} = \frac{5}{\frac{1}{3}} \]

\[ H_{\text{limit}} = 15 \]

So, the limiting height of the tree is 15 meters.