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

1 Answer
Jan 28, 2017

Answer:

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#