Examples of geometric in the following topics:
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- A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio r.
- For example, the sequence 2,6,18,54,⋯ is a geometric progression with common ratio 3.
- Similarly 10,5,2.5,1.25,⋯ is a geometric sequence with common ratio 21.
- For instance: 1,−3,9,−27,81,−243,⋯ is a geometric sequence with common ratio −3.
- The behavior of a geometric sequence depends on the value of the common ratio.
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- Geometric series are one of the simplest examples of infinite series with finite sums.
- A geometric series is an infinite series whose terms are in a geometric progression, or whose successive terms have a common ratio.
- If the terms of a geometric series approach zero, the sum of its terms will be finite.
- A geometric series with a finite sum is said to converge.
- Find the sum of the infinite geometric series 64+32+16+8+⋯
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- The formula for the sum of a geometric series can be used to convert the decimal to a fraction:
- In the case of the Koch snowflake, its area can be described with a geometric series.
- Excluding the initial 1, this series is geometric with constant ratio r=94.
- The first term of the geometric series is a=391=31, so the sum is
- Apply geometric sequences and series to different physical and mathematical topics
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- By utilizing the common ratio and the first term of a geometric sequence, we can sum its terms.
- The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant.
- The following are several geometric series with different common ratios.
- For r≠1, the sum of the first n terms of a geometric series is:
- Calculate the sum of the first n terms in a geometric sequence
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- If you add up all the terms of a geometric sequence (one in which each entry is the previous entry multiplied by a constant), you have a geometric series.
- The common example of a trend that follows a geometric series is the number of people infected with a virus, as each person passes it to several more.
- Once again, pause to convince yourself that this will work on all geometric series, but only on geometric series.
- Finally—once again—we can apply this trick to the generic geometric series to find a formula.
- So the total number of people infected follows a geometric series.
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- A geometric sequence follows the formula an=r⋅an−1. This is another example of a recursive formula.
- An applied example of a geometric sequence involves the spread of the flu virus.
- Suppose each infected person will infect two more, such that the terms follow a geometric sequence.
- Using this equation, the recursive equation for this geometric sequence is: an=2⋅an−1.
- Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence.
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- This leads to a way to visualize multiplying and dividing complex numbers geometrically.
- Sometimes it is helpful to think of complex numbers in a different geometric way.
- The previous geometric idea where the number z=a+bi is associated with the point (a,b) on the usual xy-coordinate system is called rectangular coordinates.
- This way of thinking about multiplying and dividing complex numbers gives a geometric way of thinking about those operations.
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- Two complex conjugates of each other multiply to be a real number with geometric significance.
- Note that (a+bi)(a−bi)=a2−abi+abi−b2i2=a2+b2. This number has a geometric significance.
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- Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram, three of whose vertices are O, A, and B (as shown in ).
- Addition of two complex numbers can be done geometrically by constructing a parallelogram.
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- Geometrically, z* is the "reflection" of z about the real axis (as shown in the figure below).
- Geometric representation of z and its conjugate in the complex plane.