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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{AMSsymbols} \require{color}u_{1} &= u_{1} \\u_{2} &= […]
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 […]
An arithmetic sequence is a sequence where there is a common difference between any two successive terms. $$ \large \require{AMSsymbols} \require{color} \color{red} u_{n} = u_{1}+(n-1)d$$ 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 a population of $5000$ […]
Algebraic Definition An $\textit{Arithmetic Sequence}$ is a sequence in which each term differs from the previous one by the same fixed number, 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 which: the difference between […]