 # Sum of an Infinite Geometric Series

Home > Sequences and 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. Sum to Infinity 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 [...] # Geometric Series

Home > Sequences and 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. Geometric Series Formula If we are adding the first $n$ terms of an infinite [...] # Arithmetic Series

Home > Sequences and 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. Arithmetic Series Sum of a Finite Arithmetic Series If the first term is $u_{1}$ and the common [...] # Sigma Notation

Home > Sequences and Series 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 [...] Home > Sequences and Series Suppose you invest $$2000 in the bank. The money attract an interest rate of 10% per annum. The interest is added to the investment each year, so the total interest increases. Compound Interest Problems The percentage increase each year is 10%, so at the end of the year you will [...] # Geometric Sequence Problems Home > Sequences and Series Growth and Decay Problems of growth and decay involve repeated multiplications by a constant number, common ratio. We can thus use geometric sequences to model these situations.$$\require{color} \color{red}u_{n} = u_{1} \times r^{n-1}\require{color} \color{red}u_{n+1} = u_{1} \times r^{n}$$Geometric Sequence Problems using Formula Example 1 The initial population of [...] # Geometric Sequences Home > Sequences and Series Geometric Sequence Definition Geometric Sequences are sequences where each term is obtained by multiplying the preceding term by a certain constant factor, which is often called \textit{common ratio}. A geometric sequence is also referred to as a \textit{geometric progression}. Geometric Sequences Formula David expects 10% increase per month to deposit [...] # Arithmetic Sequence Problems Home > Sequences and Series An arithmetic sequence is a sequence where there is a common difference between any two successive terms.$$\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 [...] Home > Sequences and Series 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 [...] # General Term of a Number Sequence Home > Sequences and Series 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 [...] 