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Question 1199475: The 7th term is 56 and the 12th term is -1792 of the geometric
sequence. Find the ratio and the first term. Assume the
ratios are equal.
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
r = common ratio
x = 7th term
xr = 8th term
xr^2 = 9th term
xr^3 = 10th term
xr^4 = 11th term
xr^5 = 12th term
Each time we need a new term, we multiply the previous term by the common ratio r.
The gap from the 7th term and the 12th term is 12-7 = 5, which is the exponent for xr^5.
In this case, x = 56 is the seventh term.
This means xr^5 updates to 56r^5
Set this equal to the given 12th term -1792
56r^5 = -1792
r^5 = -1792/56
r^5 = -32
From here you can use mental math to determine r = -2 because (-2)^5 = -32
Or you could say this
r^5 = -32
r = (-32)^(1/5)
r = -2
The exponent of 1/5 represents the 5th root.
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The nth term of a geometric sequence is a*r^(n-1)
a = first term
r = common ratio
n = term number
Let's plug r = -2 and n = 7.
We'll use the fact the 7th term is 56.
nth term = a*r^(n-1)
7th term = a*r^(7-1)
56 = a*(-2)^(7-1)
56 = a*64
a = 56/64
a = 7/8
In decimal form this would be exactly 0.875, but I'll stick to the fraction form.
The nth term is therefore,
a(n) = (7/8)*(-2)^(n-1)
To verify this answer, plug n = 7 and you should get a(7) = 56.
Also, you should find that a(12) = -1792.
I'll let you do these verification steps.
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Answers:
First term = 7/8
Common ratio = -2
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
The 12th term is the 7th term, multiplied by the common ratio 12-7 = 5 times. Use the given information to determine the common ratio.
-1792 = 56(r^5)
r^5 = -1792/56 = -32
r = -2
The 7th term is the first term, multiplied by the common ratio 7-1 = 6 times. Use the given 7th term and the common ratio of -2 to find the first term.
56 = a((-2)^6)
56 = 64a
a = 56/64 = 7/8
ANSWERS: common ratio -2; first term 7/8
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