How do you find the geometric means in each sequence -2, __, __, __, __, -243/16?

Jan 28, 2017

Geometric mean of the sequence is $- 5.51$

Explanation:

Here we are given a geometric sequence with $6$ terms, whose first term ${a}_{1} = - 2$ and sixth term is ${a}_{6} = - \frac{243}{16}$.

As ${n}^{t h}$ term of a geometric sequence with ${a}_{1}$ as first term and common ratio as $r$ is given by ${a}_{n} = {a}_{a} \times {r}^{\left(n - 1\right)}$

Hence in the given series ${a}_{6} = \left(- 2\right) \times {r}^{5} = - \frac{243}{16}$

i.e. ${r}^{5} = \frac{243}{16} \times \frac{1}{- 2} = \frac{243}{32} = {3}^{5} / {2}^{5}$

and hence $r = \frac{3}{2}$

as the six terms are $- 2 , - 2 r , - 2 {r}^{2} , - 2 {r}^{3} , - 2 {r}^{4} , - 2 {r}^{5}$

their geometric mean is $\sqrt[6]{- 2 \times \left(- 2 r\right) \times \left(- 2 {r}^{2}\right) \times \left(- 2 {r}^{3}\right) \times \left(- 2 {r}^{4}\right) \times \left(- 2 {r}^{5}\right)}$

= $\sqrt[6]{{\left(- 2\right)}^{6} \times {r}^{15}} = - 2 \times {r}^{\frac{15}{6}} = - 2 \times {\left(\frac{3}{2}\right)}^{\frac{5}{2}}$

= $- 2 \times 2.75568 = - 5.51136 \approx - 5.51$