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Chapter 5: Problem 73

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{20}\), when \(a_{1}=2, r=3\).

Short Answer

Expert verified
The 20th term of the given geometric sequence, \(a_{20}\), is 750599937895082.

Step by step solution

01

Identifying Parameters

Initially, it's necessary to identify the parameters. For the proposed geometric sequence, the first term (\(a_{1}\)) is 2, and the common ratio (\(r\)) is 3.
02

Applying Geometric Sequence Formula

To find the 20th value (\(a_{20}\)), apply the formula for a term in a geometric sequence, which is \(a_n = a_1 \times r^{(n-1)}\). Substituting the known values, the formula becomes: \(a_{20} = 2 \times 3^{(20-1)}\).
03

Calculate the Desired Term

Perform the calculations to obtain \(a_{20}\), which means: \(a_{20} = 2 \times 3^{19}\). Utilizing a calculator may be necessary for this calculation due to the large exponent.
04

Simplify the Result

By using a calculator, the final result simplifies to \(a_{20} = 750599937895082\). This indicates that the 20th term of this geometric sequence is 750599937895082.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-6, r=-5\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{1}{2}, 1, \frac{3}{2}, 2, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-8, r=-5\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{40}\), when \(a_{1}=6, r=-1\).

What is a geometric sequence? Give an example with your description.
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