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