Examples of geometric sequence in the following topics:
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- The $n$th term of a geometric sequence with initial value $a$ and common ratio $r$ is given by
- 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.
- For instance: $1,-3,9,-27,81,-243, \cdots$ 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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- An applied example of a geometric sequence involves the spread of the flu virus.
- Suppose each infected person will infect two more people, such that the terms follow a geometric sequence.
- Using this equation, the recursive equation for this geometric sequence is:
- Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence.
- Use a recursive formula to find specific terms of a sequence
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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.
- We can use a formula to find the sum of a finite number of terms in a sequence.
- Find the sum of the first five terms of the geometric sequence $\left(6, 18, 54, 162, \cdots \right)$.
- Calculate the sum of the first $n$ terms in a geometric sequence
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- Finite sequences are sometimes known as strings or words and infinite sequences as streams.
- Finite sequences include the empty sequence $( \quad )$ that has no elements.
- These are called recursive sequences.
- A geometric sequence is a list in which each number is generated by multiplying a constant by the previous number.
- The explicit definition of a geometric sequence is obtained in a similar way.
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- Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence.
- Then the sequence looks like:
- Then the sequence would look like:
- The second sequence of differences is:
- For example, the geometric sequence $2, 4, 8, 16,\dots$ is given by the general term $2^n$.
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- Summation is the operation of adding a sequence of numbers; the result is their sum or total.
- If you add up all the terms of an arithmetic sequence (a sequence in which every entry is the previous entry plus a constant), you have an arithmetic series.
- 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.
- Once again, pause to convince yourself that this will work on all geometric series, but only on geometric series.
- So the total number of people infected follows a geometric series.
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- Geometric series have applications in math and science and are one of the simplest examples of infinite series with finite sums.
- Geometric series are used throughout mathematics.
- 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.
- Apply geometric sequences and series to different physical and mathematical topics
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- An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
- An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
- For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic sequence with common difference of $2$.
- The behavior of the arithmetic sequence depends on the common difference $d$.
- Calculate the nth term of an arithmetic sequence and describe the properties of arithmetic sequences
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- In mathematics, a sequence is an ordered list of objects, or elements.
- Unlike a set, order matters in sequences, and exactly the same elements can appear multiple times at different positions in the sequence.
- A sequence is a discrete function.
- Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2,4,6, \cdots )$.
- Sequences of statements are necessary for mathematical induction.
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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 $64+ 32 + 16 + 8 + \cdots$