Find a formula for the general term of the sequence \{ -6,4,-\frac{8}{3}, \frac{16}{9},...

Question:

Find a formula for the general term of the sequence {eq}\{ -6,4,-\frac{8}{3}, \frac{16}{9}, -\frac{32}{27}, \cdots \} {/eq}

Geometric Sequences:

The given series can be checked for geometric series, by finding the ratio of the consecutive terms, Thus the geometric series are those series that have the common ratio, this ratio is the ratio of the consecutive terms. If the ratio is less than 1 then it forms a power series with the limiting sum as some finite value.

Answer and Explanation: 1


Lets find the ratio of the consecutive terms as follows:

{eq}\frac{4}{-6}\\ \frac{-8/3}{4}=\frac{2}{-3}\\ \frac{16/9}{-8/3}=\frac{2}{-3}\\ \frac{-32/27}{16/9}=\frac{2}{-3}\\ {/eq}

Thus this form a geometric series of the common ratio of {eq}r=\frac{2}{-3}\\ {/eq}

thus the series is {eq}\sum_{n=0}^{n} (-6)\left (\frac{2}{-3} \right )^n {/eq}


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Using Sigma Notation for the Sum of a Series

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Chapter 21 / Lesson 13
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Sigma notation can be used to present the same information as the sum of a series in a standardized manner. Explore examples of Sigma notation, and discover the way that it reduces patterns to more convenient visuals without altering the series itself.


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