Question

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.

Answer 1

6th term given is : 6
8th term given is : 216

Now use the formula:

\[\large t_n = a.r^{n-1}\]

Here put n = 6

\[\large 6 = a.r^5\]
\[\large a = \frac{6}{r^5}\]

and also:

\[\huge 216 = a.r^7\]

\[\large \frac{216}{6} = r^2\]

\[r = 6\]

Therefore,
\[\large a = \frac{6}{6^5} = 6^{-4}\]

So, 7th term is:

\[\large t_7 = ar^6 = 6^{-4} \times 6^6 = 6^2 = 36\]

Answer 2

Second Method is very easier than this:

If a, b and c are in geometric sequence then:

\[\large b = \sqrt{a \times c}\]

Use this concept there,

6, x, 216..

Now a = 6, c = 216

Therefore,
\[\large x = \sqrt{6 \times 216} = \sqrt{36 \times 36} = 36\]

So, the \(7th\) term is \(36\).