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