# Geometric Sequence Problems

## Growth and Decay

Problems of growth and decay involve repeated multiplications by a constant number, common ratio. We can thus use geometric sequences to model these situations. $$\require{color} \color{red}u_{n} = u_{1} \times r^{n-1}$$ $$\require{color} \color{red}u_{n+1} = u_{1} \times r^{n}$$

Geometric Sequence Problems using Formula

### Example 1

The initial population of chicken on a farm was $40$. The population increased by $5$% each week.

(a) How many chickens were present after $20$ weeks?
(b) How long would it take for the population to reach $240$?

### Example 2

The population of rabbits in an island at the beginning of $2018$ was $560$. The population has been steadily decreasing by $4$% per year.

(a) Find the population in the beginning of year $2022$.
(b) In which year would we expect that population to have declined to $140$?