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Chapter 7: Problem 32

Find the eighth term of a geometric sequence whose fourth term is 7 and whose fifth term is 4 .

Short Answer

Expert verified
The eighth term of the geometric sequence is approximately \(0.0748\).

Step by step solution

01

Find the common ratio

To find the common ratio, divide the fifth term by the fourth term: Common ratio (r) = Fifth term / Fourth term = 4 / 7.
02

Calculate the third term

Now that we have the common ratio, let's find the third term. Since the fourth term is 7, we can find the third term by dividing the fourth term by the common ratio: Third term = Fourth term / r = 7 / (4/7) = 7 * (7/4) = 49/4.
03

Calculate the first term

Next, let's find the first term. Using the common ratio again, we can find the first term by dividing the third term by the common ratio twice: First term = Third term / (r * r) = (49/4) / ((4/7) * (4/7)) = (49/4) * (49/16) = 2401/64.
04

Calculate the eighth term

Finally, we can find the eighth term by multiplying the first term by the common ratio raised to the power of 7: Eighth term = First term * (r^7) = (2401/64) * ((4/7)^7) = (2401/64) * (16384/823543) = 40199872/536870912. The eighth term of the geometric sequence is approximately 0.0748.

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