Geometric Sequences and Series
Geometric Sequences
A geometric sequence is an ordered list of numbers in which each term after the first is found by multiplying the previous one by a constant called [latex]r[/latex], the common ratio.Learning Objectives
Calculate the [latex]n[/latex]th term of a geometric sequence given the initial value [latex]a[/latex] and common ratio [latex]r[/latex]Key Takeaways
Key Points
- The general form of a geometric sequence is: [latex]a, ar, ar^2, ar^3, ar^4, \cdots[/latex]
- The [latex]n[/latex]th term of a geometric sequence with initial value [latex]n[/latex] and common ratio [latex]r[/latex] is given by: [latex]{ a }_{ n }=a{ r }^{ n-1 }[/latex].
Key Terms
- geometric sequence: An ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. Also known as a geometric progression.
Definition of Geometric Sequences
A geometric progression, also known as a geometric sequence, is an ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio [latex]r[/latex]. For example, the sequence [latex]2, 6, 18, 54, \cdots[/latex] is a geometric progression with common ratio [latex]3[/latex]. Similarly [latex]10,5,2.5,1.25,\cdots[/latex] is a geometric sequence with common ratio [latex]\displaystyle{\frac{1}{2}}[/latex]. Thus, the general form of a geometric sequence is: [latex-display]a, ar, ar^2, ar^3, ar^4, \cdots[/latex-display] The [latex]n[/latex]th term of a geometric sequence with initial value [latex]a[/latex] and common ratio [latex]r[/latex] is given by [latex-display]{ a }_{ n }=a{ r }^{ n-1 }[/latex-display] Such a geometric sequence also follows the recursive relation: [latex-display]a_n=ra_{n-1}[/latex-display] for every integer [latex]n\ge 1.[/latex]Behavior of Geometric Sequences
Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric series may be negative, resulting in an alternating sequence. An alternating sequence will have numbers that switch back and forth between positive and negative signs. For instance: [latex]1,-3,9,-27,81,-243, \cdots[/latex] is a geometric sequence with common ratio [latex]-3[/latex]. The behavior of a geometric sequence depends on the value of the common ratio. If the common ratio is:- Positive, the terms will all be the same sign as the initial term
- Negative, the terms will alternate between positive and negative
- Greater than [latex]1[/latex], there will be exponential growth towards positive infinity ([latex]+\infty[/latex])
- [latex]1[/latex], the progression will be a constant sequence
- Between [latex]-1[/latex] and [latex]1[/latex] but not [latex]0[/latex], there will be exponential decay toward [latex]0[/latex]
- [latex]-1[/latex], the progression is an alternating sequence (see alternating series)
- Less than [latex]-1[/latex], for the absolute values there is exponential growth toward positive and negative infinity (due to the alternating sign)
Summing the First n Terms in a Geometric Sequence
By utilizing the common ratio and the first term of a geometric sequence, we can sum its terms.Learning Objectives
Calculate the sum of the first [latex]n[/latex] terms in a geometric sequenceKey Takeaways
Key Points
- The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant.
- The general form of an infinite geometric series is: [latex]\displaystyle{\sum _{ n=0 }^{ \infty }{ { z }^{ n } }}[/latex]. The behavior of the terms depends on the common ratio [latex]r[/latex].
- For [latex]r\neq 1[/latex], the sum of the first [latex]n[/latex] terms of a geometric series is given by the formula [latex]\displaystyle{s = a\frac { 1-{ r }^{ n } }{ 1-r } }[/latex].
Key Terms
- geometric series: An infinite sequence of numbers to be added, whose terms are found by multiplying the previous term by a fixed, non-zero number called the common ratio.
- geometric progression: A series of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Infinite geometric series: Each of the purple squares is obtained by multiplying the area of the next larger square by [latex]\displaystyle{\frac{1}{4}}[/latex]. The area of the first square is [latex]\displaystyle{\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}}[/latex], and the area of the second square is [latex]\displaystyle{\frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16}}[/latex].
The following are several geometric series with different common ratios. The behavior of the terms depends on the common ratio [latex]r[/latex]:
- [latex]4+40+400+4000+\dots[/latex] has the common ratio [latex]10[/latex]
- [latex]\displaystyle{9+3+1+\frac{1}{3}+\frac{1}{9}+\dots}[/latex] has the common ratio [latex]{\frac{1}{3}}[/latex]
- [latex]3+3+3+3+\dots[/latex] has the common ratio [latex]1[/latex]
- [latex]\displaystyle{1-\frac{1}{2}+\frac{1}{4} -\frac{1}{8}+\dots}[/latex] has the common ratio [latex]-\frac{1}{2}[/latex]
- [latex]3-3+3-3+\dots[/latex] has the common ratio [latex]-1[/latex]
- If [latex]r[/latex] is between [latex]-1[/latex] and [latex]+1[/latex], the terms of the series become smaller and smaller, approaching zero in the limit, and the series converges to a sum. Consider a sequence where [latex]r[/latex] is one-half [latex]{\left(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \cdots \right)}[/latex], which has a sum of one.
- If [latex]r[/latex] is greater than [latex]1[/latex] or less than [latex]-1[/latex], the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. The series diverges.
- If [latex]r[/latex] is equal to [latex]1[/latex], all of the terms of the series are the same. The series diverges.
- If [latex]r[/latex] is [latex]-1[/latex], the terms take two values alternately [latex]\left(\text{e.g.}, 2,-2,2,-2,2,-2,\cdots \right)[/latex]. The sum of the terms oscillates between two values [latex]\left(\text{e.g.}, 2,0,2,0,2,0,\cdots \right)[/latex]. This is a different type of divergence and again the series has no sum.
Example
Find the sum of the first five terms of the geometric sequence [latex]\left(6, 18, 54, 162, \cdots \right)[/latex]. In this case, [latex]a=6[/latex] and [latex]n=5[/latex]. Also, note that [latex]r = 3[/latex], because each term is multiplied by a factor of [latex]3[/latex] to find the subsequent term. Substituting these values into the sum formula, we have: [latex]\displaystyle{ \begin{align} s &= a\frac { 1-{ r }^{ n } }{ 1-r } \\ &= 6\cdot\frac { 1-{ 3 }^{ 5 } }{ 1-3 } \\ &= 6\cdot\frac {{ -242 } }{ -2 } \\ &= 6 \cdot 121 \\ &= 726 \end{align} }[/latex]Infinite Geometric Series
Geometric series are one of the simplest examples of infinite series with finite sums.Learning Objectives
Calculate the sum of an infinite geometric series and recognize when a geometric series will convergeKey Takeaways
Key Points
- The sum of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite.
- For an infinite geometric series that converges, its sum can be calculated with the formula [latex]\displaystyle{s = \frac{a}{1-r}}[/latex].
Key Terms
- converge: Approach a finite sum.
- geometric series: An infinite sequence of summed numbers, whose terms change progressively with a common ratio.
Example
Find the sum of the infinite geometric series [latex]64+ 32 + 16 + 8 + \cdots[/latex] First, find [latex]r[/latex], or the constant ratio between each term and the one that precedes it: [latex]\displaystyle{ \begin{align} r &= \frac{32}{64} \\ &= \frac{1}{2} \end{align} }[/latex] Substitute [latex]a=64[/latex] and [latex]\displaystyle r= \frac{1}{2}[/latex] into the formula for the sum of an infinite geometric series: [latex]\displaystyle{ \begin{align} s &= \frac{64}{1-\frac{1}{2}} \\ &= \frac{64}{\frac{1}{2}} \\ &= 128 \end{align} }[/latex]Applications of Geometric Series
Geometric series have applications in math and science and are one of the simplest examples of infinite series with finite sums.Learning Objectives
Apply geometric sequences and series to different physical and mathematical topicsKey Takeaways
Key Points
- A repeating decimal can be viewed as a geometric series whose common ratio is a power of [latex]\displaystyle{\frac{1}{10}}[/latex].
- Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line.
- The interior of the Koch snowflake is a union of infinitely many triangles. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure.
- Knowledge of infinite series allows us to solve ancient problems, such as Zeno's paradoxes.
Key Terms
- geometric series: An infinite sequence of summed numbers, whose terms change progressively with a common ratio.
- fractal: A natural phenomenon or mathematical set that exhibits a repeating pattern that can be seen at every scale.
Repeating Decimal
A repeating decimal can be thought of as a geometric series whose common ratio is a power of [latex]\displaystyle{\frac{1}{10}}[/latex]. For example: [latex-display]\displaystyle{0.7777 \cdots = \frac{7}{10} + \frac{7}{100}+ \frac{7}{1000}+ \frac{7}{10000}+ \cdots}[/latex-display] The formula for the sum of a geometric series can be used to convert the decimal to a fraction: [latex]\displaystyle{ \begin{align} 0.7777 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{ 7 }{ 10 } }{ 1-\frac{ 1 }{ 10 } } \\ &= \frac{\left(\frac{7}{10}\right)}{\left(\frac{9}{10}\right)} \\ &= \left(\frac{7}{10}\right)\left(\frac{10}{9}\right)\\ &= \frac { 7 }{ 9 } \end{align} }[/latex] The formula works for any repeating term. Some more examples are: [latex]\displaystyle{ \begin{align} 0.123412341234 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{1234}{ 10000 } }{ 1-\frac{ 1 }{ 10000 } } \\ &= \frac{\left(\frac{1234}{ 10000 }\right)}{\left(\frac{9999}{10000}\right)} \\ &= \left(\frac{1234}{ 10000 }\right)\left(\frac{10000}{9999}\right)\\ &= \frac { 1234 }{ 9999 } \end{align} }[/latex] [latex]\displaystyle{ \begin{align} 0.0909090909 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{9}{ 100 } }{ 1-\frac{ 1 }{ 100 } } \\ &= \frac{\left(\frac{9}{ 100 }\right)}{\left(\frac{99}{100}\right)} \\ &= \left(\frac{9}{100}\right)\left(\frac{100}{99}\right)\\ &= \frac { 9 }{ 99 } \\ &= \frac{1}{11} \end{align} }[/latex] [latex]\displaystyle{ \begin{align} 0.143814381438 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{1438}{ 10000 } }{ 1-\frac{ 1 }{ 10000 } } \\ &= \frac{\left(\frac{1438}{ 10000 }\right)}{\left(\frac{9999}{10000}\right)} \\ &= \left(\frac{1438}{ 10000 }\right)\left(\frac{10000}{9999}\right)\\ &= \frac { 1438 }{ 9999 } \end{align} }[/latex] [latex]\displaystyle{ \begin{align} 0.9999 \cdots &= \frac { a }{ 1-r } \\ &= \frac { \frac{ 9 }{ 10 } }{ 1-\frac{ 1 }{ 10 } } \\ &= \frac{\left(\frac{9}{10}\right)}{\left(\frac{9}{10}\right)} \\ &= \left(\frac{9}{10}\right)\left(\frac{10}{9}\right)\\ &= \frac { 9 }{ 9 } \\ &= 1 \end{align} }[/latex] That is, a repeating decimal with a repeating part of length [latex]n[/latex] is equal to the quotient of the repeating part (as an integer) and [latex]10^n - 1[/latex].Archimedes' Quadrature of the Parabola
Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles. Archimedes' Theorem: Archimedes' dissection of a parabolic segment into infinitely many triangles.
[latex-display]\displaystyle{1+2 \left ( \frac { 1 }{ 8 } \right ) +4 { \left ( \frac { 1 }{ 8 } \right ) }^{ 2 }+8{ \left ( \frac { 1 }{ 8 } \right ) }^{ 3 }+ \cdots}[/latex-display]
The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives:
[latex-display]\displaystyle{1+ \frac{1}{4}+ \frac{1}{16}+ \frac{1}{64}+ \cdots}[/latex-display]
This is a geometric series with common ratio [latex]\displaystyle{\frac{1}{4}}[/latex], and the fractional part is equal to [latex]\displaystyle{\frac{1}{3}}[/latex].
Fractal Geometry
Koch snowflake: The interior of a Koch snowflake is comprised of an infinite amount of triangles.
Constructing the Koch snowflake: the first four iterations: Each iteration adds a set of triangles to the outside of the shape.
[latex-display]\displaystyle{1+3 \left ( \frac { 1 }{ 9 } \right ) +12{ \left ( \frac { 1 }{ 9 } \right ) }^{ 2 }+48{ \left ( \frac { 1 }{ 9 } \right ) }^{ 3 }+ \cdots}[/latex-display]
The first term of this series represents the area of the first triangle, the second term the total area of the three triangles added in the second iteration, the third term the total area of the twelve triangles added in the third iteration, and so forth. Excluding the initial term [latex]1[/latex], this series is geometric with constant ratio [latex]\displaystyle{r = \frac{4}{9}}[/latex]. The first term of the geometric series is [latex]\displaystyle{a = 3 \frac{1}{9} = \frac{1}{3}}[/latex], so the sum is:
[latex]\displaystyle{
\begin{align}
1+ \frac { a }{ 1-r } &=1+ \frac { \frac{1}{3} }{ 1- \frac{4}{9} } \\
&= \frac{8}{5}
\end{align}
}[/latex]
Thus the Koch snowflake has [latex]\displaystyle{\frac{8}{5}}[/latex] of the area of the base triangle.
Zeno's Paradoxes
Zeno's Paradoxes are a set of philosophical problems devised by an ancient Greek philosopher to support the doctrine that the truth is contrary to one's senses. Simply stated, one of Zeno's paradoxes says: There is a point, A, that wants to move to another point, B. If A only moves half of the distance between it and point B at a time, it will never get there, because you can continue to divide the remaining space in half forever. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. We now know that his paradox is not true, as evidenced by the convergence of the geometric series with [latex]\displaystyle{r = \frac{1}{2}}[/latex]. This problem has been solved by modern mathematics, which can apply the concept of infinite series to find a sum of the distances traveled.Licenses & Attributions
CC licensed content, Shared previously
- Curation and Revision. Authored by: Boundless.com. License: Public Domain: No Known Copyright.
CC licensed content, Specific attribution
- Geometric sequence. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- IB Mathematics (HL)/Algebra. Provided by: Wikibooks License: CC BY-SA: Attribution-ShareAlike.
- geometric sequence. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- Geometric sequence. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- Geometric series. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- geometric series. Provided by: Wiktionary License: CC BY-SA: Attribution-ShareAlike.
- Geometric Series. Provided by: Wikipedia License: Public Domain: No Known Copyright.
- Geometric series. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- Geometric progression. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- Infinite series. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- geometric series. Provided by: Wiktionary License: CC BY-SA: Attribution-ShareAlike.
- Geometric Series. Provided by: Wikipedia License: Public Domain: No Known Copyright.
- Zeno's paradoxes. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- Geometric progression. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- Geometric series. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- annuity. Provided by: Wiktionary License: CC BY-SA: Attribution-ShareAlike.
- fractal. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- geometric series. Provided by: Wiktionary License: CC BY-SA: Attribution-ShareAlike.
- Geometric Series. Provided by: Wikipedia License: Public Domain: No Known Copyright.
- Koch snowflake. Provided by: Wikipedia License: CC BY-SA: Attribution-ShareAlike.
- Geometric series. Provided by: Wikipedia Located at: https://en.wikipedia.org/wiki/Geometric_series. License: Public Domain: No Known Copyright.
- Koch similarity tiling. Provided by: Wikipedia License: Public Domain: No Known Copyright.