# Arithmetic Series

Home 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. Arithmetic Series Sum of a Finite Arithmetic Series If the first term is $u_{1}$ and the common difference is $d$, the [...]

# Sigma Notation

Home 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 [...]

# Compound Interest

Home Suppose you invest $$2000$ in the bank. The money attract an interest rate of $10$% per annum. The interest is added to the investment each year, so the total interest increases. Compound Interest Problems The percentage increase each year is $10$%, so at the end of the year you will have $110$% of the [...]

# Geometric Sequence Problems

Home 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 [...]

# Geometric Sequences

Home 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 $\textit{geometric progression}$. Geometric Sequences Formula David expects $10$% increase per month to deposit to his account. A [...]

# Arithmetic Sequence Problems

Home 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. Arithmetic Sequence Problems Example 1 A city is studies and found to have [...]

# Arithmetic Sequences

Home Algebraic Definition An $\textit{Arithmetic Sequence}$ is a sequence in which each term differs from the previous one by the same fixed number, which is often called $\textit{common difference}$. It can also be referred to as an $\textit{arithmetic progression}$. A sequence in mathematics is an ordered set of numbers. An $\textit{arithmetic sequence}$ is one in [...]

# Exponential Growth and Decay using Logarithms

Home It has been known that how exponential functions can be used to model a variety of growth and decay situations. These included the growth of populations and the decay of radioactive substances. In this lesson we consider more growth and decay problems, focusing particularly on how logarithms can be used in there solution. Exponential [...]

# General Term of a Number Sequence

Home Sequences may be defined in one of the following ways: listing all terms of a finite sequence: $2, 5, 8, 11, 14, 17$ listing the first few terms and assuming that the pattern represented continuous indefinitely: $2, 5, 8, \cdots $ giving a description in words: $\textit{Starts at 2, and each term is 3 [...]

# Number Sequences

Home In $\textit{sequences}$, it is important that we can; recognise a pattern in a set of numbers describe the pattern in words continue the pattern. A $\textit{number sequence}$ is an ordered list of numbers defined by a rule. The sequence starts at $4$ and add $5$ each time. $4, 9, 14, 19, \cdots$ The numbers [...]

# Graphing Natural Logarithmic Functions

Home The inverse function of $y=e^x$ is $y=\log_{e}{x}$. Therefore $y=\log_{e}{x}$ is an inverse function, it is a reflection of $y=e^x$ in the line $y=x$. The graphs of $y=e^x$ is $y=\log_{e}{x}$: \begin{array}{|c|c|c|} \require{color} \hline & y=e^x & \color{red}y =\log_{e}{x} \\ \hline \text{domain} & x \in \mathbb{R} & \color{red}x \gt 0 \\ \hline \text{range} & y \gt [...]

# Graphing Logarithmic Functions

Home The inverse function of $y=a^x$ is $y=\log_{a}{x}$. Therefore $y=\log_{a}{x}$ is an inverse function, it is a reflection of $y=a^x$ in the line $y=x$. The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $0 \lt a \lt 1$: The graphs of $y=a^x$ is $y=\log_{a}{x}$ for $a \gt 1$: \begin{array}{|c|c|c|} \require{color} \hline & y=a^x & \color{red}y =\log_{a}{x} \\ [...]

# Logarithm Change of Base Rule

Home $$\log_{b}{a} = \dfrac{\log_{c}{a}}{\log_{c}{b}}$$ $$\text{for }a,b,c>0 \text{ and } b,c \ne 1$$ For example, \( \begin{align} \log_{3}{8} &= \dfrac{\log_{2}{8}}{\log_{2}{3}} \\ &= \dfrac{\log_{5}{8}}{\log_{5}{3}} \\ &= \dfrac{\log_{10}{8}}{\log_{10}{3}} \\ &\vdots \\ &= 1.8927 \cdots \\ \end{align} \) $\textit{Proof:}$ \( \begin{align} \displaystyle \text{Let } \log_{b}{a} &= x \cdots (1)\\ b^x &= a \\ \log_{c}{b^x} &= \log_{c}{a} &\text{taking logarithm in [...]

# Exponential Inequalities using Logarithms

Home Inequalities are worked in exactly the same way except that there is a change of sign when dividing or multiplying both sides of the inequality by a negative number. \begin{array}{|c|c|c|} \hline \log_{2}{3}=1.6>0 & \log_{5}{3}=0.7>0 & \log_{10}{3}=0.5>0 \\ \hline \log_{2}{2}=1>0 & \log_{5}{2}=0.4>0 & \log_{10}{2}=0.3>0 \\ \hline \log_{2}{1}=0 & \log_{5}{1}=0 & \log_{10}{1}=0 \\ \hline \log_{2}{0.5}=-1 \log_{10}{9} [...]

# Exponential Equations using Logarithms

Home We can find solutions to simple exponential equations where we could make equal bases and then equate exponents (indices). For example, $2^{x}=8$ can be written as $2^x = 2^3$. Therefore the solution is $x=3$. However, it is not always easy to make the bases the same such as $2^x=5$. In these situations we use [...]

# When No Need to Apply Quotient Rule for Differentiating a Fraction

The following examples show that you do not need to apply the quotient rule for differentiating when the denominator is a constant. Please see the following cases with the same question. 1. $\textit{Application of the quotient rule}$ \( \begin{align} \displaystyle \dfrac{d}{dx}\dfrac{48x – 4x^3}{3} &= \frac{{\frac{d}{{dx}}\left( {48x – 4{x^3}} \right) \times 3 – \left( {48x – […]

# Logarithmic Equations

Home We can use the laws of logarithms to write equations in a different form. This can be particularly useful if an unknown appears as an index (exponent). $$2^x=7$$ For the logarithmic function, for every value of $y$, there is only one corresponding value of $x$. $$y=5^x$$ We can therefore take the logarithm of both [...]

# Natural Logarithm Laws

Home The laws for natural logarithms are the laws for logarithms written in base $e$: $$ \begin{align} \displaystyle \ln{x} + \ln{y} &= \ln{(xy)} \\ \ln{x} - \ln{y} &= \ln{\dfrac{x}{y}} \\ \ln{x^n} &= n\ln{x} \\ \ln{e} &= 1 \\ \end{align} $$ Note that $\ln{x}=\log_{e}{x}$ and $x>0,y>0$. Example 1 Use the laws of logarithms to write $\ln{4} [...]

# Natural Logarithms

Home After $\pi$, the next weird number is called $e$, for $\textit{exponential}$. It was first discussed by Jacob Bernoulli in 1683. It occurs in problems about compound interest, leds to logarithms, and tells us how variables like radioactivity, temperature, or the human population increase or decrease. In 1614 John Napier knew, from personal experience, that [...]

# Logarithmic Laws

Home $$ \log_{a}{(xy)} = \log_{a}{x} + \log_{a}{y} $$ $\textit{Proof}$ Let $A=\log_{a}{x}$ and $B=\log_{a}{y}$. Then $a^A = x$ and $a^B=y$. \( \begin{align} a^A \times a^B&= xy \\ a^{A+B} &= xy \\ A+B &= \log_{a}{(xy)} \\ \therefore \log_{a}{x}+\log_{a}{y} &= \log_{a}{(xy)} \\ \end{align} \) $$\log_{a}{\dfrac{x}{y}} = \log_{a}{x} - \log_{a}{y} $$ $\textit{Proof}$ Let $A=\log_{a}{x}$ and $B=\log_{a}{y}$. Then $a^A = [...]