SOLUTION: If the first 3 items of a geometric progression are a − 1, a +3, and 3 a +1 for some positive number a, what is the numerical value of the fourth term?

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Question 957592: If the first 3 items of a geometric progression are a − 1, a +3, and 3 a +1 for some positive
number a, what is the numerical value of the fourth term?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given:
the first 3 terms of a geometric progression are
a-1,
a+%2B3, and
3a+%2B1
for some positive number a
In a Geometric Sequence each term is found by multiplying the previous term by a constant.
Let a constant be k:
%28a+-1%29k=+a+%2B3
k=+%28a+%2B3%29%2F%28a+-1%29..........(1)
%28a+%2B3%29k=+3a+%2B1
k=+%283a+%2B1%29%2F%28a+%2B3%29..........(2)
make %281%29=%282%29
%28a+%2B3%29%2F%28a+-1%29=%283a+%2B1%29%2F%28a+%2B3%29 ............cross multiply
%28a+%2B3%29%28a+%2B3%29=%283a+%2B1%29%28a+-1%29
a%5E2+%2B3a+%2B3a%2B9=3a%5E2-3a+%2Ba+-1
a%5E2+%2B6a%2B9=3a%5E2-2a+-1
0=3a%5E2-2a+-1-a%5E2+-6a-9
2a%5E2-8a+-10+=0........simplify
a%5E2-4a+-5=0
a%5E2%2Ba-5a+-5=0
%28a%5E2%2Ba%29-%285a+%2B5%29=0
a%28a%2B1%29-5%28a+%2B1%29=0
%28a-5%29%28a+%2B1%29=0
solutions:
a=5
or
a=-1
since given that first 3 terms are a+-1, a+%2B3, and 3a+%2B1 for some positive
number a, our solution is a=5
so, your terms are:
a+-1=5-1=4,
a+%2B3=5%2B8=8, and
3a+%2B1=3%2A5%2B1=16
now find k

k=+%28a+%2B3%29%2F%28a+-1%29..........(1)
k=+8%2F4
k=+2
so, the fourth term is 2%2A16=32