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?

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Answer 1 · 2015-11-19T07:21:05

Find that #a_7 = sqrt(a_6a_8)#, hence #a_7 = 36#

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.

So #a_6 = a r^5#, #a_7 = a r^6# and #a_8 = a r^7#

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)#

That is: #a_7 = sqrt(a_6 a_8)#

In other words, #a_7# is the geometric mean of #a_6# and #a_8#

In our particular example #a_6 = 6# and #a_8 = 216 = 6^3#,

So:

#a_7 = sqrt(a_6 a_8) = sqrt(6*6^3) = sqrt(6^2*6^2) = 6^2 = 36#