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?

1 Answer
Mar 23, 2016

Answer:

If the common ratio is positive, then:

#a_7 = sqrt(a_6 * a_8)#

Explanation:

If you are given the #6th# and #8th# terms of a geometric series then there are two possibilities for the #7th# term, namely #+-sqrt(a_6 * a_8)#

If you are told that the common ratio is positive or that all of the terms of the sequence are positive then #a_7 = sqrt(a_6 * a_8)# is the geometric mean of #a_6# and #a_8#. Otherwise it could be #-sqrt(a_6 * a_8)#.

Why does this work?

The general term of a geometric sequence can be written:

#a_n = a r^(n-1)#

where #a# is the initial term and #r# the common ratio.

If #a, r > 0# then:

#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)#

#color(white)()#
In the given example #a_6 = 12# and let us suppose #a_8 = 48#

Then #a_7 = sqrt(a_6 * a_8) = sqrt(12 * 48) = sqrt(24*24) = 24#