SOLUTION: 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.
Algebra.Com
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.
---------------------------------------------------------
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.
---------------------------------------------------------
Answers:
First term = 7/8
Common ratio = -2
Answer by greenestamps(13200) (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
RELATED QUESTIONS
The seventh term is 56 and the 12th term is -1792 of the geometric progression. Find the... (answered by josgarithmetic)
Find the first term and the common difference of the arithmetic sequence whose
7th term (answered by ikleyn)
If the 4th term of a geometric sequence is 15 and the 7th term is 405, find the common... (answered by CubeyThePenguin)
Find the 12th term of a Geometric Sequence whose 3rd term is 432 and the 5th term is... (answered by MathLover1,greenestamps,ikleyn)
The first term of a geometric sequence is 300 and the common ratio is 0.6. What is the... (answered by ewatrrr)
The fifth term of a geometric sequence is 15 and the eighth term is -15/8. Find the first (answered by mananth)
The 12th term of an arithmetic sequence is 35. The 7th term is four times the 2nd term.... (answered by richwmiller)
Find the first term & common difference if:
10th term is 35;15th terms is 55
2nd term... (answered by ikleyn)
If the 4th term of an arithmetic sequence is 24 and the 12th term is 56, what is the... (answered by josgarithmetic)