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 \( 2053 \).
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses