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What's next after 1,\frac{1}{2},\frac{1}{4},\frac{1}{8}........ ?

Question:

What's next after {eq}1,\frac{1}{2},\frac{1}{4},\frac{1}{8}........ {/eq}?

Geometric Series

The geometric progression is the sequence of the numbers and each value or term is obtained by mutinying the previous term with the common ratio. The common ratio is the ratio of the term and its previous term in a geometric progression.

Answer and Explanation:


Given Data

  • The given series is: {eq}1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},........ {/eq}


The general series of a GP is,

{eq}a,ar,a{r^2},a{r^3}........ {/eq}


Here, the common ratio is {eq}r {/eq} and the first term is {eq}a {/eq}.


Compare the standard series with the given series.

{eq}\begin{align*} a &= 1\\ r &= \dfrac{1}{2} \end{align*} {/eq}


The next terms can be calculated as,

{eq}n = \dfrac{1}{8}\left( r \right) {/eq}


Substitute the known values,

{eq}\begin{align*} n &= \dfrac{1}{8}\left( {\dfrac{1}{2}} \right)\\ &= \dfrac{1}{{16}} \end{align*} {/eq}


Thus, the next term is {eq}\dfrac{1}{{16}} {/eq}.


Learn more about this topic:

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How to Calculate a Geometric Series

from Math 101: College Algebra

Chapter 12 / Lesson 6
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