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Arithmetic Series

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

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 [...]
Geometric Sequence Problems

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

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

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

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 [...]
Logarithm Change of Base Rule

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

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} [...]
When No Need to Apply Quotient Rule for Differentiating a Fraction

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 – […]

Natural Logarithms

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

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 = [...]