Please solve and show work.  Find the sixth term of a geometric sequence  with t5 = 24 and t8 = 3.

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You need to use the formula that helps you to find each term of the geometric series such that:

`t_n = t_1*q^(n-1)`

`t_1`  represents the first term of geometric series

q represents the common ratio of geometric series

Reasoning by analogy yields:

`t_5 = t_1*q^4 => 24 = t_1*q^4`

`t_8 = t_1*q^7 => 3 = t_1*q^7 => 3 = t_1*q^(4+3)=> 3 = t_1*q^4*q^3`

Substituting 24 for `t_1*q^4`  yields:

`3 = 24*q^3 => 3/24 = q^3 => q = root(3)(1/8) => q = 1/2`

`24 = t_1*(1/2)^4 => 24 = t_1*1/16 => t_1 = 24*16 => t_1 = 384`

Since you know the values of `t_1`  and q, you may evaluate `t_6`  such that:

`t_6 = t_1*q^5 => t_6 = 384*(1/32) => t_6 = 12`

Hence, evaluating the sixth term of the given geometric series yields `t_6 = 12` .

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