How do you use the geometric mean to find the 7th term in a geometric sequence if the 6th term is 12 and the 8th term?
If the common ratio is positive, then:
#a_7 = sqrt(a_6 * a_8)#
If you are given the
If you are told that the common ratio is positive or that all of the terms of the sequence are positive then
Why does this work?
The general term of a geometric sequence can be written:
#a_n = a r^(n-1)#
#sqrt(a_6 * a_8) = sqrt(ar^5 * a r^7) = sqrt(ar^6 * ar^6) = ar^6 = a_7#
In fact, in general:
#sqrt(a_n * a_(n+2)) = a_(n+1)#
In the given example