The 1st term of a geometric sequence is 3 and the eighth term is 384. Find the common ratio, the sum and the product of the firs
1 answer:
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Answer:
- ratio: 2
- sum: 765
- product: 1,761,205,026,816
Step-by-step explanation:
The general term of a geometric sequence is ...
an = a1×r^(n-1)
Here, we're given that the first term is a1 = 3. From the 8th term, we can find the common ratio, r.
a8 = 384 = 3r^7
128 = r^7 . . . . . . . . divide by 3
2 = r . . . . . . . . . 7th root
The common ratio is 2.
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The sum of the terms of a geometric series is ...
Sn = a1×(r^n -1)/(r -1)
For the known values of a1 and r, the sum o 8 terms is ...
S8 = 3×(2^8 -1)/(2 -1) = 3(255) = 765
The sum of the first 8 terms is 765.
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The product of the first 8 terms will be ...
(3)(3×2)(3×2^2)(3×2^3)...(3×2^6)(3×2^7) = (3^8)(2^(1+2+3+...+7))
= (3^8)(2^28) = 6561 × 268,435,456
Many calculators cannot report the full value of this product. (Many spreadsheets can.)
The product of the first 8 terms is 1,761,205,026,816.
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In the attached, we found the value of r that made the 8th term be 384 by solving the equation 3×r^7 -384 = 0. The value is shown on the graph as the x-intercept of the exponential function. That x-intercept is 2, so r=2 (as above).
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