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Growth by a fixed fraction

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A position of size $81$ grows by exactly one third of its current size in every period. What is its size after $3$ periods?

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Hint

A gain of one third means the multiplier is a simple fraction.

Solution

The multiplier per period is $1 + \frac{1}{3} = \frac{4}{3}$ .
After $3$ periods the size is $81 \times (\frac{4}{3})^{3} = 81 \times \frac{64}{27}$ .
Cancel: $81 \div 27 = 3$ .
The size is $3 \times 64 = 192$ .

Answer: $192$