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 […]

# Sequences and Series

# Minimum Loan Repayment and Number of Months of Loan Repayment

Jane borrows \( \$100 \ 000 \) which is to be repaid in equal monthly instalments. The interest rate is \( 6 \% \) per annum reducible, compounded monthly. It can be shown that the amount, \( \$b_n \), owing after the \( n \) th repayment is given by the formula: $$ b_n = […]

# Sum of an Infinite Geometric Series

To examine the sum of all the terms of an infinite geometric sequence, we need to consider $S_n = \dfrac{u_1(1-r^n)}{1-r}$ when $n$ gets very large. If $\left|r\right|>1$, the series is said to be divergent and the sum infinitely large.For instance, when $r=2$ and $u_1=1$;$S_\infty=1+2+4+8+\cdots$ is infinitely large. If $\left|r\right|<1$, or $-1 \lt r \lt 1$, […]

# Geometric Series

A $\textit{geometric series}$ is the sum of the terms of a geometric sequence.for example: $1, 2, 4, 8, \cdots , 2048$ is a finite geometric sequence. $1+2+4+8+ \cdots +2048$ is the corresponding finite geometric series. If we are adding the first $n$ terms of an infinite geometric sequence, we are then calculating a finite geometric […]

# Arithmetic Series

An $\textit{arithmetic series}$ is the sum of the terms of an arithmetic sequence.For example: $4, 7, 10, 13, \cdots,61$ is a finite arithmetic sequence. $4+7+10+13+ \cdots +61$ is the corresponding arithmetic series. If the first term is $u_{1}$ and the common difference is $d$, the terms are:$$u_{1},u_{1}+d,u_{1}+2d,u_{1}+3d,\cdots$$\( \begin{align} \displaystyle \require{color}u_{1} &= u_{1} \\u_{2} &= u_{1} […]

# Sigma Notation

Another mathematical device that is widely used in sequences and series is called $\textit{sigma notation}$. The Greek letter, $\sum$ (capital sigma), is used to indicate the sum of a sequence. For example:$$\sum_{n=1}^{10}{n^2} = 1^2 + 2^2 + 3^2 + \cdots + 10^2$$The limits of the sum, the numbers on the bottom and top of the […]

# Compound Interest of Investment

Suppose you invest $\$2000$ in the bank. The money attracts an interest rate of $10\%$ per annum. The interest is added to the investment each year, so the total interest increases. The percentage increase each year is $10\%$, so at the end of the year, you will have $110\%$ of the value at its start. […]

# Geometric Sequence Problems

Problems of growth and decay involve repeated multiplications by a constant number, common ratio. We can thus use geometric sequences to model these situations. $$ \begin{align} \displaystyle\require{color} \color{red}u_{n} &= u_{1} \times r^{n-1} \\\require{color} \color{red}u_{n+1} &= u_{1} \times r^{n}\end{align}$$ Example 1 The initial population of chicken on a farm was $40$. The population increased by $5$% […]

# Geometric Sequence and Geometric Mean

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 […]

# Arithmetic Sequence Problems

An arithmetic sequence is a sequence where there is a common difference between any two successive terms. $$\require{color} \color{red} u_{n} = u_{1}+(n-1)d$$ where $\require{color} \color{red} u_{1}$ is the first term and $\require{color} \color{red}d$ is the common difference of the arithmetic sequence. Example 1 A city studies and found to have a population of $5000$ in […]