If the fifth and sixth terms of a geometric sequence are 8 and 16 respectively, then what is the third term?

1 Answer
SCooke
Jun 27, 2018

#x_3 = 2#

Explanation:

In a Geometric Sequence each term is found by multiplying the previous term by a constant.
In General we write a Geometric Sequence like this:
#{a, ar, ar^2, ar^3, ... }#
where:
a is the first term, and
r is the factor between the terms (called the "common ratio")
We can also calculate any term using the Rule:
#x_n = ar^(n-1)#
(We use "n-1" because #ar^0# is for the 1st term)

#x_5 = 8 = ar^(4)#
#x_6 = 16 = ar^(5)#
#x_6/x_5 = 16/8 = 2 = ar^(5)/ar^(4) = r#
#8 = a2^(4)# ; #a = 8/16 = 0.5#,

SO the series is #x_n = 0.5xx2^(n-1)#

CHECK: #16 = 0.5xx2^(5) = 0.5(32) = 16#

FOR #n = 3#: #x_3 = 0.5xx2^(2) = 0.5xx4 = 2#

We could continue that for the whole series.
#x_1 = 0.5#
#x_2 = 1.0#
#x_3 = 2#
#x_4 = 4#
#x_5 = 8#
#x_6 = 16#
https://www.mathsisfun.com/algebra/sequences-sums-geometric.html

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