# Geometric Sequence Problems

Growth and decay problems involve repeated multiplications by a constant number, a common ratio. We can thus use geometric sequences to model these situations.

$$ \large \begin{align} \require{AMSsymbols} \displaystyle

\require{color} \color{red}u_{n} &= u_{1} \times r^{n-1} \\

\require{color} \color{red}u_{n+1} &= u_{1} \times r^{n}

\end{align}$$

## Example 1

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

(a) How many chickens were present after $20$ weeks?

\( \begin{align} \displaystyle

u_{1} &= 40 \\

r &= 1 + 0.05 = 1.05 \\

u_{n} &= 40 \times 1.05^{n-1} \\

u_{20} &= 40 \times 1.05^{20-1} \\

&= 101.078 \cdots

\end{align} \)

There were $101$ chickens.

(b) How long would it take for the population to reach $240$?

\( \begin{align} \displaystyle

40 \times 1.05^{n-1} &= 240 \\

1.05^{n-1} &= 6 \\

n-1 &= \log_{1.05}{6} \\

n &= \log_{1.05}{6} + 1 \\

&= \dfrac{\log_{10}{6}}{\log_{10}{1.05}}\ + 1 \\

&= 37.723 \cdots

\end{align} \)

The population will reach $240$ in the \( 38 \)^{th} week.

## Example 2

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

(a) Find the population at the beginning of the year \( 2022 \).

\( \begin{align} \displaystyle

u_{1} &= 560 &2018\\

r &= 1-0.04 = 0.96 \\

u_{4} &= 560 \times 0.96^3 &2022 \\

&= 495.45 \cdots

\end{align} \)

The rabbit population will be approximately \( 495 \) at the beginning of \( 2022 \).

(b) In which year would we expect that population to have declined to $140$?

\( \begin{align} \displaystyle

560 \times 0.96^{n-1} &= 140 \\

0.96^{n-1} &= 140 \div 560 \\

&= 0.25 \\

n-1 &= \log_{0.96}{0.25} \\

n &= \log_{0.96}{0.25} + 1 \\

&= \dfrac{\log_{10}{0.25}}{\log_{10}{0.96}} + 1 \\

&= 34.959 \cdots \\

\end{align} \)

The rabbits’ population will be approximately \(1405 \) at \( 35 \) years.

So it will be \( 2053 \).

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