Question

Insert four geometric means between -8 and 1/4​

Answer 1

Answer:

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Answer 2

Given,

First terms of geometric series $\[ = - 8\]$

Last term of geometric series $\[ = \frac{1}{4}\]$

Solution,

Last term will be 6th term.

$\[\frac{1}{4} = a{r^{6 - 1}}\]$

$\[ \Rightarrow \frac{1}{4} = \left( { - 8} \right){r^5}\]$

$\[ \Rightarrow {r^5} = - \frac{1}{{32}}\]$

$\[ \Rightarrow r = - \frac{1}{2}\]$

First geometric means,

$\[\begin{array}{l} = ar\\ = \left( { - 8} \right) \times \left( { - \frac{1}{2}} \right)\\ = 4\end{array}\]$

Second geometric means,

$\[\begin{array}{l} = a{r^2}\\ = \left( { - 8} \right) \times {\left( { - \frac{1}{2}} \right)^2}\\ = - 2\end{array}\]$

Third geometric means,

$\[\begin{array}{l} = a{r^3}\\ = \left( { - 8} \right) \times {\left( { - \frac{1}{2}} \right)^3}\\ = 1\end{array}\]$

Four geometric means,

$\[\begin{array}{l} = a{r^4}\\ = \left( { - 8} \right) \times {\left( { - \frac{1}{2}} \right)^4}\\ = - \frac{1}{2}\end{array}\]$

Hence the four geometric means are $\[4, - 2,1, - \frac{1}{2}\]$