Algebra 2 - Geometric Sequence & Series

Introduction

All about Geometric Sequences

• A geometric sequence (aka geometric progression) is a special type of sequence in which the ratio between two consecutive terms is always the same (constant).
• The ratio is called common ratio $\mathit{\left(}\mathit{r}\mathit{\right)}$.
• Some examples of geometric sequences are given in the figure below.

• Based on the number of terms, geometric sequences are classified into two types - finite and infinite.
• The first sequence shown in the figure above represents a finite geometric sequence (with $\mathit{r}\mathbf{=}\mathbf{2}$) as it has a finite number of terms.
• On the other hand, the second sequence (with $\mathit{r}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$) has an infinite number of terms (indicated by dots) and hence is an example of an infinite geometric sequence.
• In general, a geometric sequence is represented as: ${\mathit{a}}_{\mathbf{1}}\mathbf{,}\mathbf{ }{\mathit{a}}_{\mathbf{1}}\mathit{r}\mathbf{,}\mathbf{ }{\mathit{a}}_{\mathbf{1}}{\mathit{r}}^{\mathbf{2}}\mathbf{,}\mathbf{ }{\mathit{a}}_{\mathbf{1}}{\mathit{r}}^{\mathbf{3}}\mathbf{,}\mathbf{ }{\mathit{a}}_{\mathbf{1}}{\mathit{r}}^{\mathbf{4}}\mathbf{,}\mathbf{ }\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}$, where ${\mathit{a}}_{\mathbf{1}}\mathbf{=}$first term and $\mathit{r}\mathbf{=}$common ratio.
• The ${\mathit{n}}^{\mathbf{t}\mathbf{h}}$ term of a geometric sequence is, ${\mathit{a}}_{\mathbf{n}}\mathbf{=}{\mathit{a}}_{\mathbf{1}}{\mathit{r}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$.

 

All about Geometric Series

• It is the sum of the terms of a geometric sequence.
• The formula to find the sum of '$\mathit{n}$' terms of a finite geometric sequence is ${\mathit{S}}_{\mathbf{n}}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{1}}\mathbf{·}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{r}}^{\mathbf{n}}\mathbf{)}}{(1-r)}$, provided $r\ne 1$.
• The formula to find the sum of '$\mathit{n}$' terms of an infinite geometric sequence with $\mathbf{\left|}\mathbf{r}\mathbf{\right|}\mathbf{<}\mathbf{1}$ is ${\mathit{S}}_{\mathbf{\infty }}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{1}}}{(1-r)}$.

Solved Examples

Example 1: Is $\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{10}\mathbf{,}\mathbf{ }\mathbf{15}\mathbf{,}\mathbf{ }\mathbf{20}\mathbf{,}\mathbf{ }\mathbf{25}$ a geometric sequence?

Solution: The condition for a sequence to be a geometric sequence is that it must have a common ratio. Since this condition is not satisfied for the given sequence, it is not a geometric sequence.

[Note: In the given sequence, the difference between any two consecutive terms is the same (common difference $\mathit{d}\mathbf{=}\mathbf{5}$) and hence is an arithmetic sequence.]

 

Example 2: Find the first six terms of a geometric sequence in which the first term is ${\mathit{a}}_{\mathbf{1}}\mathbf{=}\mathbf{3}$ and the common ratio is $\mathit{r}\mathbf{=}\mathbf{2}$.

Solution: It is given that the first term is ${\mathit{a}}_{\mathbf{1}}\mathbf{=}\mathbf{3}$ and the common ratio is $\mathit{r}\mathbf{=}\mathbf{2}$. We know that the ${\mathit{n}}^{\mathbf{t}\mathbf{h}}$ of a a geometric sequence is ${\mathit{a}}_{\mathbf{n}}\mathbf{=}{\mathit{a}}_{\mathbf{1}}{\mathit{r}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$. We can find the other terms using this formula.

Second term, $a_2=\left(3\right)·{\left(2\right)}^{2-1}$$=\left(3\right)·{\left(2\right)}^1=6$

Third term, $a_3=\left(3\right)·{\left(2\right)}^{3-1}$$=\left(3\right)·{\left(2\right)}^2=12$

Fourth term, $a_4=\left(3\right)·{\left(2\right)}^{4-1}$$=\left(3\right)·{\left(2\right)}^3=24$

Fifth term, $a_5=\left(3\right)·{\left(2\right)}^{5-1}$$=\left(3\right)·{\left(2\right)}^4=48$

Sixth term, $a_6=\left(3\right)·{\left(2\right)}^{6-1}$$=\left(3\right)·{\left(2\right)}^5=96$

Therefore, the first six terms of the given geometric sequence are $\mathbf{3}\mathbf{,}\mathbf{ }$$\mathbf{6}\mathbf{,}\mathbf{ }$$\mathbf{12}\mathbf{,}\mathbf{ }$$\mathbf{24}\mathbf{,}\mathbf{ }$$\mathbf{48}\mathbf{,}$ and$\mathbf{ }\mathbf{96}$.

 

Example 3: Find the next term of a geometric sequence $\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{1}\mathbf{,}\mathbf{ }\frac{\mathbf{1}}{\mathbf{5}}\mathbf{,}\mathbf{ }\frac{\mathbf{1}}{\mathbf{25}}\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}$

Solution: Here, the first four terms of the geometric sequence are given. The first term is, ${\mathit{a}}_{\mathbf{1}}\mathbf{=}\mathbf{5}$. We can calculate the common ratio, $\mathit{r}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}$.

Now, the next term means the fifth term i.e. $a_5=\left(5\right)·{\left(\frac{1}{5}\right)}^{5-1}$$=\left(5\right)·{\left(\frac{1}{5}\right)}^4$$=\left(5\right)·\left(\frac{1}{5\times 5\times 5\times 5}\right)$$=\frac{1}{125}$

 

Example 4: Find the sum of the first ten terms of the given geometric sequence: $\mathbf{2}\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{6}\mathbf{,}\mathbf{ }\mathbf{18}\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{54}\mathbf{,}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}$

Solution: Here, the first term ${\mathit{a}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$, common ratio $\mathit{r}\mathbf{=}\frac{\mathbf{-}\mathbf{6}}{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{3}$. We know that, $S_n=\frac{a_1·\left(1-r^n\right)}{\left(1-r\right)}$.

$S_{10}=\frac{\left(2\right)·\left[1-{\left(-3\right)}^{10}\right]}{\left[1-\left(-3\right)\right]}$$=\frac{\left(2\right)·\left(1-59049\right)}{\left(1+3\right)}$$=\frac{\left(2\right)·\left(-59048\right)}{4}$$=-29524$

 

Example 5: Find the sum of the infinite geometric series $\mathbf{27}\mathbf{,}\mathbf{ }$$\mathbf{9}\mathbf{,}\mathbf{ }$$\mathbf{3}\mathbf{,}\mathbf{ }$$\mathbf{1}\mathbf{,}\mathbf{ }$$\frac{\mathbf{1}}{\mathbf{3}}\mathbf{,}\mathbf{ }$$\frac{\mathbf{1}}{\mathbf{9}}\mathbf{,}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }\mathbf{.}\mathbf{ }$

Solution: Here, the first term ${\mathit{a}}_{\mathbf{1}}\mathbf{=}\mathbf{27}$, common ratio $\mathit{d}\mathbf{=}\frac{\mathbf{9}}{\mathbf{27}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}$. We know that for $\left|r\right|<1$, ${\mathit{S}}_{\mathbf{\infty }}\mathbf{=}\frac{\mathbf{a}}{\left(1-r\right)}$.

$S_{\infty }=\frac{27}{(1-{\displaystyle \frac{1}{3}})}$$=\frac{27}{\left({\displaystyle \frac{2}{3}}\right)}$$=\frac{81}{2}$

Cheat Sheet

• The common ratio of any geometric sequence is found by dividing any term of the geometric sequence by its preceding term i.e. $\mathit{r}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{n}}}{{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}}$
• ${\mathit{n}}^{\mathbf{t}\mathbf{h}}$ term of a geometric sequence from the beginning: ${\mathit{a}}_{\mathbf{n}}\mathbf{=}{\mathit{a}}_{\mathbf{1}}\mathbf{·}{\mathit{r}}^{\mathbf{n}\mathbf{-}\mathbf{1}}$
• ${\mathit{m}}^{\mathbf{t}\mathbf{h}}$ term of a finite geometric sequence (having '$\mathit{n}$' terms) from the end: ${\mathit{a}}_m\mathbf{=}{\mathit{a}}_{\mathbf{1}}\mathbf{·}{\mathit{r}}^{\mathbf{n}\mathbf{-}\mathit{m}}$
• For a finite geometric sequence, the sum of the first '$\mathit{n}$' terms: ${\mathit{S}}_{\mathbf{n}}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{1}}\mathbf{·}\mathbf{(}\mathbf{1}\mathbf{-}{\mathbf{r}}^{\mathbf{n}}\mathbf{)}}{(1-r)}$, where $\mathit{r}\mathbf{\ne }\mathbf{1}$.
• For $\mathit{r}\mathbf{=}\mathbf{1}$, the sum of the first '$\mathit{n}$' terms of the finite geometric sequence: ${\mathit{S}}_{\mathbf{n}}\mathbf{=}\mathit{n}{\mathit{a}}_{\mathbf{1}}$.
• For an infinite geometric sequence where $\mathbf{\left|}\mathbf{r}\mathbf{\right|}\mathbf{<}\mathbf{1}$, the sum of first '$\mathit{n}$' terms: ${\mathit{S}}_{\mathbf{\infty }}\mathbf{=}\frac{{\mathbf{a}}_{\mathbf{1}}}{(1-r)}$.
• For an infinite geometric sequence where $\mathbf{\left|}\mathbf{r}\mathbf{\right|}\mathbf{\ge }\mathbf{1}$, the sum cannot be computed.
• If $\mathit{x}\mathbf{,}\mathbf{ }\mathit{y}\mathbf{,}\mathbf{ }\mathit{z}$ represents a set (group) of three consecutive terms of any geometric sequence, then $\mathit{y}$ is called the geometric mean of the set, and its value is $\mathit{y}\mathbf{=}\sqrt{\mathbf{x}\mathbf{z}}$.

Blunder Areas

• When each term of a geometric sequence is multiplied by a non-zero number, the resulting sequence is also a geometric sequence with the same common ratio.
• The new sequence obtained by taking the reciprocal of terms of a geometric sequence is also a geometric sequence.
• If all the terms of a given geometric sequence are raised to the same power, then the resulting sequence is also a geometric sequence.