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A ball is dropped from a height of \( 16 \) metres onto a timber floor and bounces. After each bounce, the maximum height reached by the ball is \( 80 \% \) of the previous maximum height. Thus, after it first hits the floor, it reaches a height of \( 12.8 \) metres before […]
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 […]
Geometric Sequence Definition Geometric Sequences are sequences where each term is obtained by multiplying the preceding term by a certain constant factor, which is often called $\textit{common ratio}$. A geometric sequence is also referred to as a $\textit{geometric progression}$. David expects $10$% increase per month to deposit to his account. A $10$% increase per month […]
A geometric sequence is also referred to as a geometric progression. Each term of a geometric sequence can be obtained from the previous one by multiplying by the same non-zero constant. For example, \(2, \ 6, \ 18, \ 54, \cdots \) is a geometric sequence as each term can be obtained by multiplying the […]
Applications of Geometric Sequence using Logarithms Geometric Sequence using Logarithms is used to find the number of geometric sequences terms. Question 1 Find the largest \(5\)-digit term of the sequence \( 2, 4, 8, 16, \cdots \). \( \begin{aligned} \displaystyle2 \times 2^{n-1} &\lt 100000 \\2^{n} &\lt 100000 \\n &\lt \log_{2}{100000} \\n &\lt 16.609 \cdots \\n […]