Geometric Series

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 [...]
Arithmetic Series

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. Sum of a Finite Arithmetic Series If the first term is $u_{1}$ and the common difference is $d$, the terms are: $$u_{1},u_{1}+d,u_{1}+2d,u_{1}+3d,\cdots$$ [...]
Sigma Notation

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 [...]
Geometric Sequence Problems

Geometric Sequence Problems

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. $$ \begin{align} \displaystyle \require{color} \color{red}u_{n} &= u_{1} \times r^{n-1} \\ \require{color} \color{red}u_{n+1} &= u_{1} \times r^{n} \end{align}$$ Example 1 The initial population of chicken on a farm was [...]
Geometric Sequences

Geometric Sequences

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 Sequences

Arithmetic Sequences

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 which: [...]