Question

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

Answer 1

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=-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`