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

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Answer 1 · 2017-01-28T03:25:23

Geometric mean of the sequence is $-5.51$

Here we are given a geometric sequence with $6$ terms, whose first term $a_1=-2$ and sixth term is $a_6=-243/16$ .

As $n^(th)$ term of a geometric sequence with $a_1$ as first term and common ratio as $r$ is given by $a_n=a_axxr^((n-1))$

Hence in the given series $a_6=(-2)xxr^5=-243/16$

i.e. $r^5=243/16xx1/(-2)=243/32=3^5/2^5$

and hence $r=3/2$

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

their geometric mean is $root(6)(-2xx(-2r)xx(-2r^2)xx(-2r^3)xx(-2r^4)xx(-2r^5))$

= $root(6)((-2)^6xxr^15)=-2xxr^(15/6)=-2xx(3/2)^(5/2)$

= $-2xx2.75568=-5.51136~~-5.51$

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