How do you use the geometric mean to find the 7th term in a geometric sequence if the 6th term is 6 and the 8th term is 216?
The general term of a geometric sequence can be written:
#a_n = a r^(n-1)#
So we find:
#a_7 = a r^6 = sqrt(a r^6 * a r^6) = sqrt(a r^5 * a r^7) = sqrt(a_6 a_8)#
In other words,
In our particular example
#a_7 = sqrt(a_6 a_8) = sqrt(6*6^3) = sqrt(6^2*6^2) = 6^2 = 36#